College Trigonometry, Sixth Edition

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College Trigonometry

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College Trigonometry SIXTH EDITION

Richard N. Aufmann Vernon C. Barker Richard D. Nation Palomar College

H O U G H T O N M I F F L I N C O M PA N Y Boston New York

Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Senior Marketing Manager: Jennifer Jones Marketing Associate: Mary Legere Associate Editor: Noel Kamm Editorial Associate: Andrew Lipsett Senior Project Editor: Carol Merrigan Editorial Assistant: Anthony D’Aries Art and Design Manager: Gary Crespo Cover Design Manager: Anne S. Katzeff Photo Editor: Jennifer Meyer Dare Composition Buyer: Chuck Dutton

Cover photograph: © Dan Hill/cityofsound.com

PHOTO CREDITS Chapter 1: p. 1: John Foxx/Stockbyte Silver/Getty Images. Chapter 2: p. 117: Andrew Brookes/CORBIS; p. 128: Bryan Mullennix/Iconica/Getty Images; p. 132: Courtesy of NASA and JPL; p. 145: Tony Craddock/Getty Images; p. 159: Reuters/New Media Inc./CORBIS. Chapter 3: p. 254: Courtesy of Richard Nation. Chapter 4: p. 307: McDuff/Everton/CORBIS. Chapter 5: p. 334: Dennis De Mars/Fractal Domains/www.fractaldomains.com; p. 334: Steve Allen/Alamy; p. 351: The Granger Collection. Chapter 6: p. 371: 1998 International Conference on Quality Control by Artificial Vision – QCAV ’98, Kagawa Convention Center, Takamatsu, Kagawa, Japan, November 10–12, 1998, pp. 521–528; p. 372: Ian Morison/Jodrell Bank Conservatory; p. 374: Courtesy of Michael Levin, Opti-Gone International. Reprinted by permission; p. 389: Hugh Rooney/Eye Ubiquitous/CORBIS; p. 397: The Granger Collection. Chapter 7: p. 451: AP/Wide World Photos; p. 458: Bettmann/CORBIS; p. 463: Charles O’Rear/CORBIS; p. 464: David James/Getty Images; p. 467: Bettmann/CORBIS; p. 474: Jan Halaska/Index Stock Imagery/Jupiter Images; p. 513: Tom Brakefield/CORBIS; p. 521: Bettmann/CORBIS; p. 525: AP/Wide World.

Copyright © 2008 Houghton Mifflin Company. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photographing and recording, or by any information storage or retrieval system without the prior written permission of Houghton Mifflin Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2006933003 ISBN 10: 0-618-82507-X ISBN 13: 978-0-618-82507-3 123456789-WC-10 09 08 07

Contents Preface

1

ix

Functions and Graphs 1 1.1 O A G N R A ITS D P H S EN IQ TEquations O EA U N A IS D L and Inequalities 2 1.2 O A G N R A ITS D P H S EN IQ TA O EU N A IS D L Two-Dimensional Coordinate System and Graphs 1.3 Introduction to Functions 30 1.4 Properties of Graphs 53 1.5 The Algebra of Functions 69 1.6 Inverse Functions 81 1.7 Modeling Data Using Regression 95 Exploring Concepts with Technology: Graphing Piecewise Functions with a Graphing Calculator

15

108

Chapter 1 Summary 110 Chapter 1 Assessing Concepts 111 Chapter 1 Review Exercises 112 Quantitative Reasoning: Public Key Cryptography 114 Chapter 1 Test 116

2

Trigonometric Functions 117 2.1 lgAngles n A and Arcs 118 2.2 Right Triangle Trigonometry 134 2.3 Trigonometric Functions of Any Angle 147 2.4 Trigonometric Functions of Real Numbers 155 2.5 Graphs of the Sine and Cosine Functions 169 2.6 Graphs of the Other Trigonometric Functions 180 2.7 Graphing Techniques 192 2.8 Harmonic Motion—An Application of the Sine and Cosine Functions

203

Exploring Concepts with Technology: Sinusoidal Families 210 Chapter 2 Summary 210 Chapter 2 Assessing Concepts 212 Chapter 2 Review Exercises 212 Quantitative Reasoning: Find the Periods of Trigonometric Functions and Combined Musical Sound Tracks 214 Chapter 2 Test 214 Cumulative Review Exercises 215

v

vi

Contents

3

Trigonometric Identities and Equations 216 3.1 rVerification eV of Trigonometric Identities 217 3.2 Sum, Difference, and Cofunction Identities 225 3.3 Double- and Half-Angle Identities 236 3.4 Identities Involving the Sum of Trigonometric Functions 3.5 Inverse Trigonometric Functions 255 3.6 Trigonometric Equations 268

246

Exploring Concepts with Technology: Approximate an Inverse Trigonometric Function with Polynomials Chapter 3 Summary 284 Chapter 3 Assessing Concepts 286 Chapter 3 Review Exercises 286 Quantitative Reasoning: Basketball and Trigonometric Equations 288 Chapter 3 Test 289 Cumulative Review Exercises 290

4

Applications of Trigonometry 292 4.1 ehThe T Law of Sines 293 4.2 The Law of Cosines and Area 4.3 Vectors 312

302

Exploring Concepts with Technology: Optimal Branching of Arteries 328 Chapter 4 Summary 328 Chapter 4 Assessing Concepts 329 Chapter 4 Review Exercises 329 Quantitative Reasoning: Trigonometry and Great Circle Routes 330 Chapter 4 Test 332 Cumulative Review Exercises 332

5

Complex Numbers 334 5.1 ehComplex T Numbers 335 5.2 Trigonometric Form of Complex Numbers 5.3 De Moivre’s Theorem 350 Exploring Concepts with Technology: The Mandelbrot Iteration Procedure

343

356

Chapter 5 Summary 357 Chapter 5 Assessing Concepts 358 Chapter 5 Review Exercises 358 Quantitative Reasoning: Graphing the Mandelbrot Set 359

283

Contents Chapter 5 Test 360 Cumulative Review Exercises

6

361

Topics in Analytic Geometry 362 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Parabolas 363 Ellipses 374 Hyperbolas 390 Rotation of Axes 403 Introduction to Polar Coordinates 412 Polar Equations of the Conics 426 Parametric Equations 432

Exploring Concepts with Technology: Using a Graphing Calculator to Find the n th Roots of z

443

Chapter 6 Summary 444 Chapter 6 Assessing Concepts 445 Chapter 6 Review Exercises 446 Quantitative Reasoning: The Mathematics of a Rotary Engine 447 Chapter 6 Test 449 Cumulative Review Exercises 450

7

Exponential and Logarithmic Functions 451 7.1 7.2 7.3 7.4 7.5 7.6

Exponential Functions and Their Applications 452 Logarithmic Functions and Their Applications 467 Properties of Logarithms and Logarithmic Scales 480 Exponential and Logarithmic Equations 494 Exponential Growth and Decay 506 Modeling Data with Exponential and Logarithmic Functions

Exploring Concepts with Technology: Using a Semilog Graph to Model Exponential Decay Chapter 7 Summary 537 Chapter 7 Assessing Concepts 538 Chapter 7 Review Exercises 539 Quantitative Reasoning: Sales 541 Chapter 7 Test 542 Cumulative Review Exercises 543

Solutions to the Try Exercises S1 Answers to Selected Exercises A1 Index I1

535

521

vii

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Preface College Trigonometry, Sixth Edition builds on the strong pedagogical features of the previous edition. Always with an eye toward supporting student success, we have increased our emphasis on conceptual understanding, quantitative reasoning, and applications.

Applications We have retained our basic philosophy, which is to deliver a comprehensive and mathematically sound treatment of the topics considered essential for a college algebra course. To help students master these concepts, we have maintained a dynamic balance among theory, application, modeling, and drill. Carefully developed mathematics is complemented by abundant, relevant, and contemporary applications, many of which feature real data and tables, graphs, and charts. Ever mindful of the motivating influence that contemporary and appropriate applications have on students, we have included many new application exercises from a wide range of disciplines, and in new formats. For example, a new Quantitative Reasoning feature is found at the end of each chapter. Students are urged to investigate concepts and apply those concepts to a variety of contexts while testing the reasonableness of their answers. Applications require students to use problem-solving strategies and newly learned skills to solve practical problems, demonstrating the value of algebra. Many application exercises are accompanied by a diagram that helps students visualize the mathematics of the application.

Technology Technology is introduced very naturally to illustrate or enhance conceptual understanding of appropriate topics. We integrate technology into a discussion when it can be used to foster and promote better understanding of a concept. The optional graphing calculator exercises, optional Integrating Technology boxes, and optional Exploring Concepts with Technology features are designed to instill in students an appreciation for both the power and limitations of technology. Optional Modeling sections, which use real data, rely heavily on the use of a graphing calculator, and serve to motivate students, are incorporated throughout the text.

Aufmann Interactive Method (AIM) By incorporating many interactive learning techniques, including the key features outlined below, College Trigonometry helps students to understand concepts, work independently, and obtain greater mathematical fluency. ■

Try Exercise references follow all worked examples. This feature encourages students to test their understanding by working an exercise similar to the worked example. An icon and a page reference are given below the example, making it easy for the student to navigate to the suggested exercise and back. The complete solution to the Try Exercise can be found in the Solutions to the Try Exercises appendix. This interaction among the examples, the Try Exercises, the Solutions to the Try Exercises, and the exercise sets serves as a checkpoint for students as they read the text, do their homework, and study a section.

ix

x

Preface ■

Annotated Examples are provided throughout each section, and are titled so that students can see at a glance the type of problem being illustrated. The annotated steps assist the student in moving from step to step, and help explain the solution.

■

Question/Answer In each section, we pose at least one question that encourages the reader to pause and think about the current discussion. To ensure that the student does not miss important information, the answer to the question is provided as a footnote on the same page.

CHANGES IN THE SIXTH EDITION Overall changes ■

NEW! Quantitative Reasoning After each set of Chapter Review Exercises, a new Quantitative Reasoning scenario uses concepts from the chapter to explore an application in more depth or extend a mathematical concept from the chapter.

■

NEW! Most definitions are now immediately followed by an example, to enhance conceptual understanding.

■

NEW! A Calculus Connection icon alerts students to a connection between the current topic and calculus. This feature identifies topics that will be revisited in a subsequent calculus course or other advanced course.

■

Revised! Review Notes, which help students recognize the prerequisite skills needed to understand new concepts, are featured more prominently throughout the text, encouraging students to use them more frequently. These example-specific notes direct students to the appropriate page(s) for review, thus decreasing student frustration and creating more opportunities for comprehension.

■

Revised! We have thoroughly reviewed each exercise set. In addition to updating and adding contemporary applications, we have focused our revisions on providing a smooth progression from routine exercises to exercises that are more challenging.

■

Revised! New chapter openers demonstrate how the mathematics developed in each chapter is applied.

■

Moved! Prepare for This Section exercises, formerly called Prepare for the Next Section exercises, have been moved from the end of the section to the beginning. An up-front review gives students a chance to test their understanding of prerequisite skills and concepts before proceeding to a new topic.

■

Revised! Assessing Concepts exercises, found at the end of each chapter, have been enhanced with more question types, including fill-in-the blank, multiple choice, matching, and true/false.

■

Revised! We have highlighted more of the important points within the body of the text, to enhance conceptual understanding.

■

Enhanced! Technology program

Preface

xi

Changes in each chapter In addition to updating and adding new examples, applications, and exercises throughout, we have made a number of chapter-specific changes. Here are some of them: Chapter 1: Functions and Graphs ■ Added concepts involving the solution of literal equations ■

Added an introductory discussion of asymptotes

Chapter 2: Trigonometric Functions ■ Added exercises that involve finding a trigonometric function that can be used to model an application ■

Rewrote the material on the linear and angular speed of a point moving on a circular path

Chapter 3: Trigonometric Identities and Equations ■ Expanded the coverage concerning the verification of trigonometric identities ■

Expanded the coverage concerning the use of power-reducing identities

Chapter 4: Applications of Trigonometry ■ Added exercises that use the graph of a vector to find its components ■

Added exercises that involve the equilibrium of forces

Chapter 5: Complex Numbers ■ Included introductory material that involves the use of complex numbers and iteration to produce fractal images ■

Increased the coverage concerning the Mandelbrot iteration procedure and the Mandelbrot set

Chapter 6: Topics in Analytic Geometry ■ Included new figures to illustrate concepts involving the standard form of the equation of a parabola and its graph ■

Included new exercises that involve matching the graph of a conic section with its equation

■

Included a new example that illustrates a technique that can be used to write some polar equations in rectangular form

■

Increased the parametric equation coverage concerning the use of time as a parameter and the simulation of motion

Chapter 7: Exponential and Logarithmic Functions ■ Added exercises that use translations and/or reflections to graph exponential functions ■

Added exercises that involve the evaluation of a logarithm, without using a calculator

xii

Preface ■

Added exercises that involve finding the domain of a logarithmic function

■

Included a proof of the Product Property of logarithms. The proof of the Quotient Property and the Power Property of logarithms are given as exercises.

■

Provided additional coverage and additional exercises involving the expanding and condensing of logarithmic expressions

■

Added application exercises that can be solved by using exponential or logarithmic equations

■

Included guidelines for selecting the type of mathematical function that models a given application

ACKNOWLEDGEMENTS We would especially like to thank the users of the previous edition for their helpful suggestions on improving the text. Also, we sincerely appreciate the time, effort, and suggestions of the reviewers of this edition. Jan Archibald – Ventura College, CA Steve Armstrong – LeTourneau University, TX Rick Bailey – Midlands Technical College, SC Sandy Dieckman – Northeast Community College, NE Robert Gardner – East Tennessee State University, TN Jim Hendrickson – Indiana University, IN Masoud F. Kazemi - Lewis-Clark State College, ID Linda Kuroski – Erie Community College, NY Betty Larson – South Dakota State University, SD Helen Medley – Kent State University, OH Scott Metcalf – Eastern Kentucky University, KY Suellen Robinson – North Shore Community College, MA Gary Wardall – University of Wisconsin, Green Bay, WI

xiii

Preface

»

College Trigonometry, Sixth Edition is designed to enhance conceptual understanding and quantitative reasoning through its motivating opening features, its Interactive Method (AIM), its features for student success, its exercises, its contemporary applications, and its use of technology.

Enhance Conceptual Understanding and Quantitative Reasoning: Using Motivating Features

4

Revised! Chapter Openers Elizabeth City

New Chapter Openers demonstrate how the mathematics developed in each chapter is applied.

4.1 Nags Head

North Carolina

10

at the bottom of the page let students know of additional resources.

New Bern 39.4° Morehead City

Prepare for This Section Each section opens with review exercises, titled Prepare for This Section, which gives students a chance to test their understanding of prerequisite skills and concepts before proceeding to a new topic. An outline of the section’s contents is also provided in the margin as a study aid.

Section 4.2 I I I

The Law of Cosines Area of a Triangle Heron’s Formula

4.2

The Law of Sines The Law of Cosines and

4.3

Vectors

Trigonometry and Indir ect Measurement

In Chapter 2 we used trigonometric functions to find the unknown length of a side of a given right triang le. In this chapter we develop theorems that can be used to find the length of a side or the measure of an angle of any triang le, even if it is not a Hurricane right triangle. These theor ems are often used in the areas of navigation, surveying, and building design. Meter ologists use these theorems to estim ate the distance from an appro aching hurricane to cities in the projected path of the hurri cane. For instance, in the diagram on the left, the distance from the hurricane to Nags Head can be deter mined Sines, a theorem prese using the Law of nted in this chapter. See Exercises 30 and 31 on page 299 for addit ional applications that solved by using the Law can be of Sines.

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

The Law of Cosines and Area

SSG

Online Study Cent er For online student resourc es, such as section quizzes, visit this website: college.hmco.com/info/aufm annCAT

P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A20.

292

PS1. Evaluate 兹a2 ⫹ b2 ⫺ 2ab cos C for a 苷 10.0, b 苷 15.0, and C 苷 110.0⬚. Round your result to the nearest tenth. [2.3] PS2. Find the area of a triangle with a base of 6 inches and a height of 8.5 inches. PS3. Solve c 2 苷 a2 ⫹ b2 ⫺ 2ab cos C for C. [3.5] PS4. The semiperimeter of a triangle is defined as one-half the perimeter of the triangle. Find the semiperimeter of a triangle with sides of 6 meters, 9 meters, and 10 meters. PS5. Evaluate 兹s共s ⫺ a兲共s ⫺ b兲共s ⫺ c兲 for a 苷 3, b 苷 4, c 苷 5, and a⫹b⫹c s苷 . 2

c

PS6. State a relationship between the lengths a, b, and c in the triangle shown at the right. [1.2]

NEW! A Calculus Connection icon

兰

Calculus Connection

b

a

兰

Calculus Connection

identifies topics that will be revisited in a subsequent calculus course.

Area

64.9°

Online Study Center

ile s

, and

m

SSG ,

5

The icons

Applications of Trigonometry

Each Chapter Opener ends with a reference to a particular exercise within the chapter that asks the student to solve a problem related to the chapter opener topic.

30. NAVAL MANEUVERS The distance between an aircraft carrier and a Navy destroyer is 7620 feet. The angle of elevation from the destroyer to a helicopter is 77.2°, and the angle of elevation from the aircraft carrier to the helicopter is 59.0°. The helicopter is in the same vertical plane as the two ships, as shown in the following figure. Use this data to determine the distance x from the helicopter to the aircraft carrier.

Helicopter

x 59.0° Aircraft carrier

77.2°

7620 feet

Navy destroyer

31. CHOOSING A GOLF STRATEGY The following diagram shows two ways to play a golf hole. One is to hit the ball down the fairway on your first shot and then hit an approach shot to the green on your second shot. A second way is to hit directly toward the pin. Due to the water hazard, this is a more risky strategy. The distance AB is 165 yards, BC is 155 yards, and angle A 苷 42.0⬚. Find the distance AC from the tee directly to the pin. Assume that angle B is an obtuse angle.

The Difference Quotient The expression f共x ⫹ h兲 ⫺ f共x兲 , h

C

h苷0

is called the difference quotient of f. It enables us to study the manner in which a function changes in value as the independent variable changes.

B

A

xiv

Preface

»

Enhance Conceptual Understanding and Quantitative Reasoning: Using Contemporary Applications

NEW! Quantitative Reasoning After each set of Chapter Review Exercises, a new Quantitative Reasoning scenario explores an application in more depth or to extend a mathematical concept from the chapter.

»»» Quantitative Reason ing:

take note

Functions and Graphs

Chapter 1

92

s, of a winter selling price S, in dollar to determine the retail of x dollars. paid a wholesale price coat for which she has for the wholesale price of $96 a paid ant merch a. The price determine the retail selling winter coat. Use S to coat. she will charge for this whole ant’s ⫺1 to determine the merch b. Find S and use it retails at $399. sale price for a coat that

35. f 共x兲苷 ⫺2x ⫹ 5 36. f 共x兲苷 ⫺x ⫹ 3 2x 1 37. f 共x兲苷 x ⫺ 1 , x 苷 x 2 38. f 共x兲苷 x ⫺ 2 , x 苷

FASHION The function s共x兲苷 2x ⫹ 24 can be used to convert a U.S. women’s shoe size into function size. Determine the an Italian women’s shoe n’s shoe to convert an Italian wome s⫺1共x兲 that can be used size. shoe size to its equivalent U.S. rts a K共x兲苷 1.3x ⫺ 4.7 conve FASHION The function a54. United States to the equiv men’s shoe size in the functhe mine Deter om. d Kingd lent shoe size in the Unite d Kingdom ⫺1 used to convert a Unite tion K 共x兲 that can be equivalent U.S. shoe size. men’s shoe size to its in monthly earnings E共s兲, COMPENSATION The by 55. sales executive is given dollars, of a software s, of s is the value, in dollar where 2500, ⫹ 0.05s 苷 E共s兲 the month. the executive during the software sold by could use ⫺1 in how the executive Find E 共s兲 and expla this function. lass postage rate POSTAGE Does the first-c 56. have an inverse funcfunction given below r. tion? Explain your answe

»»»

53.

x⫺1 ⫺1 39. f 共x兲苷 x ⫹ 1 , x 苷 2x ⫺ 1 苷 ⫺3 40. f 共x兲苷 x ⫹ 3 , x 2 ⱖ0 41. f 共x兲苷 x ⫹ 1, x 2 ⱖ0 42. f 共x兲苷 x ⫺ 4, x

xⱖ2 43. f 共x兲苷兹x ⫺ 2 , xⱕ4 44. f 共x兲苷兹4 ⫺ x , 2 x ⱖ ⫺2 45. f 共x兲苷 x ⫹ 4x , 2 xⱕ3 46. f 共x兲苷 x ⫺ 6x , 2 1 , x ⱕ ⫺2 47. f 共x兲苷 x ⫹ 4x ⫺ 2 1, x ⱖ 3 48. f 共x兲苷 x ⫺ 6x ⫹

by e of a cube is given GEOMETRY The volum 3 the measure of the length V共x兲苷 x , where x is ⫺1 and explain cube. Find V 共x兲 of a side of the what it represents. function f 共x兲苷⫺112x conUNIT CONVERSIONS The and 50. s, f 共x兲. Find f 共x兲 verts feet, x, into inche ines. determ it what in expla

49.

Weight (in ounces)

The function 51. FAHRENHEIT TO CELSIUS f 共x兲苷

52.

To encrypt a message means to use a secret code to chang e the message so that it canno t be understood by an unaut horized user. To decrypt a messa ge means to change a coded messa ge back to its original form.

Public Key Cryptography

As mentioned in the Chapter Opener, perfo rming financial trans the Internet requires secur actions over e transmissions between receiver. One method two sites, the sender and of creating secure trans the missions is to use a function. modular A modular function is one that gives, in integ er form, the remainder number is divided by when one another. We write a ¢ b mod m to mean that remainder when b is divid a is the ed by m. Here are some examples. 4 ¢ 22 mod 6 because 22 ⫼ 6 苷 3 rema inder 4. 1 ¢ 37 mod 4 because 37 ⫼ 4 苷 9 rema inder 1. 0 ¢ 55 mod 11 because 55 ⫼ 11 苷 5 rema inder 0. 17 ¢ 17 mod 31 becau se 17 ⫼ 31 苷 0 remainder 17. QR1. Find the value of each expression. a. 15 mod 4 b. 37 mod 5 c. 52 mod 321 Public key cryptography uses a modular function person’s name or credi to encrypt a message— t card number—so that say, a only the receiver of the decrypt it. The message message can is decrypted by using the inverse of the modu that was used to encry lar function pt the message. Inver se functions were discu Section 1.6. ssed in

alent es Fahrenheit to an equiv is used to convert x degre ⫺1 is used. f and explain how it Celsius temperature. Find function ng merchant uses the RETAIL SALES A clothi S共x兲苷

3 x ⫹ 18 2

Carefully developed mathematics is complemented by abundant, relevant, and contemporary applications, many of which feature real data and tables, graphs, and charts. Note that applications using real data are identified

Cost

0⬍wⱕ1 1⬍wⱕ2

$.39

2⬍wⱕ3 3⬍wⱕ4

$.87

$.63 $1.11

em called the M A famous probl 57. THE BIRTHDAY PROBLE is a ranlike this: Suppose there birthday problem goes What is the of n people in a room. ay domly selected group of the people have a birthd two least at that probability you that for year? It may surprise on the same day of the two of least at that bility the proba a group of 23 people, following ay is about 50.7%. The the people share a birthd ay probabiliestimate shared birthd graph can be used to 60. ties for 1 ⱕ n ⱕ

5 共x ⫺ 32兲 9

Updated! Applications

by

.

Applications demonstrate to students the value of algebra and cover topics from a wide variety of disciplines— including agriculture, business, chemistry, construction, earth sciences, economics, education, manufacturing, medicine, nutrition, real estate, and sociology. Besides providing motivation to study mathematics, applications assist students in developing good problem-solving skills.

Projects MEDIAN–MEDIAN LINE Another linear model of data is called the median–median line. This line employs summary points calculated using the medians of subsets of the independent and dependent variables. The median of a data set is the middle number or the average of the two middle numbers for a data set arranged in numerical order. For instance, to find the median of 兵8, 12, 6, 7, 9其, first arrange the data in numerical order.

x

y

2

3

3

5

4

4

5

7

6, 7, 8, 9, 12

7

9

The median is 8, the number in the middle. To find the median of 兵15, 12, 20, 9, 13, 10其, arrange the numbers in numerical order.

8

12

9

12

10

14

6

9, 10, 12, 13, 15, 20 The median is 12.5, the average of the two middle numbers. 12 ⫹ 13 Median 苷苷 12.5 2 The median – median line is determined by dividing a data set into three equal groups. (If the set cannot be divided into three equal groups, the first and third groups should be equal. For instance, if there are 11 data points, divide the set into groups of 4, 3, and 4.) The slope of the median – median line is the slope of the line through the x-medians and y-medians of the first and third sets of points. The median – median line passes through the average of the x-and y-medians of all three sets. A graphing calculator can be used to find the equation of

Projects 16

Regression line

8

11

15

12

14

14

0 0

Median-median line

3. Consider the data set 兵共1, 3兲, 共2, 5兲, 共3, 7兲, 共4, 9兲, 共5, 11兲, 共6, 13兲, 共7, 15兲, 共8, 17兲其. a. Find the equation of the linear regression line for these data. b. Find the equation of the median – median line for these data. c.

What conclusion might you draw from the answers to a. and b.?

Projects are designed to engage the student in mathematics. At the end of each section, students are asked to do one or more of the following types of projects: –solve a more involved application problem –investigate a concept in greater depth –write a proof of a statement With some projects, students are asked to chronicle the procedure used to solve it, and to suggest an extension to the project. These projects are ideal candidates for small group assignments.

xv

Preface

»

Enhance Conceptual Understanding and Quantitative Reasoning: Using Technology Exploring Concepts with Technology

328

Chapter 4

Optional Exploring Concepts with Technology problems extend ideas from the chapter, encouraging students to use calculators or computers to investigate solutions to computationally unpleasant problems. In this way, calculators and computers have expanded the limits of the types of problems that can be solved at this level. In addition, students are challenged to think about the pitfalls of computational solutions.

Applications of Trigonometry

Exploring Concepts with Technology Optimal Branching of Arteries The physiologist Jean Louis Poiseuille (1799–1869) developed several laws concerning the flow of blood. One of his laws states that the resistance R of a blood vessel of length l and radius r is given by

R苷k

r2

P1

r1

P3

冉

R苷k

θ

a

Figure 4.41

(1)

The number k is a variation constant that depends on the viscosity of the blood. Figure 4.41 shows a large artery with radius r1 and a smaller artery with radius r2 . The branching angle between the arteries is ␪. Make use of Poiseuille’s Law, Equation (1), to show that the resistance R of the blood along the path P1 P2 P3 is b

P2

l r4

冊

a ⫺ b cot u b csc u ⫹ (r1)4 (r2)4

(2)

Use a graphing utility to graph R with k 苷 0.0563, a 苷 8 centimeters, b 苷 4 centi3 meters, r1 苷 0.4 centimeter, and r2 苷 r1 苷 0.3 centimeter. Then estimate (to the 4

nearest degree) the angle ␪ that minimizes R. By using calculus, it can be demonstrated that R is minimized when

cos ␪ 苷

冉冊 r2 r1

4

(3)

This equation is remarkable because it is much simpler than Equation (2) and because it does not involve the distance a or b. Solve Equation (3) for ␪, with 3 r2 苷 r1 . How does this value of ␪ compare with the value of ␪ you obtained 4 by graphing?

90

Chapter 1

Functions

Integrating Technology

and Graphs

Integrating Te

chnology

Using optional Integrating Technology boxes, technology is integrated into a discussion when it can be used to foster and promote a better conceptual understanding of a concept. Additionally, optional graphing calculator

Some graph ing utilities can be used function wi to draw the thout the use graph of the r Figure 1.94 inverse of a shows the gra having to find the inv erse ph of f共x 兲苷 both shown 0.1x 3 ⫺ 4. Th function. For instance, in Figure 1.95, e graphs of of f ⫺1 is the along with f and f ⫺1 are the graph of reflection of the graph of y 苷 x. Note The display that the gra f with respe shown in Fig ph ct to the gra ure 1.95 wa Plus graph ph of y 苷 x. s produced ing calculato on a TI-83/ r by using the DRAW menu TI-83 Plus/T DrawInv com . I-84 mand, which is in the 10

examples and exercises (identified by are presented throughout the text.

10

)

y=x

−15 15 f(x) = 0.1x 3

f −1 −15

−4

15 f(x) = 0.1x 3

−10

Figure 1.94

−10

Figure 1.95

Topics for Discussion

1. If f共x 兲

3

Linear Regression Models The data in the table below show the population of selected states and the number of professional sports teams (Major League Baseball, National Football League, National Basketball Association, Women’s National Basketball Association, National Hockey League) in those states. A scatter diagram of the data is shown in Figure 1.96 on page 96.

Number of Professional Sports Teams for Selected States State

Arizona California Colorado

Populations (in millions)

Number of Teams

State

Populations (in millions)

Number of Teams

5.9

5

Minnesota

5.1

36.1

17

New Jersey

8.7

3

19.3

10

9.7

3

4.7

4

Florida

17.8

11

Illinois

12.8

5

Pennsylvania

12.4

7

Indiana

6.3

3

Texas

22.9

9

10.1

5

Wisconsin

5.5

3

Michigan

New York

5

North Carolina

Modeling Special modeling sections, which rely heavily on the use of a graphing calculator, are incorporated throughout the text. These optional sections introduce the idea of a mathematical model, using various real-world data sets that further motivate students and help them see the relevance of mathematical concepts.

−4

xvi

Preface

»

Enhance Conceptual Understanding and Quantitative Reasoning: Using the Aufmann Interactive Method (AIM) By incorporating many interactive learning techniques, including the key features outlined below, College Trigonometry uses the proven Aufmann Interactive Method (AIM) to help students understand concepts, work independently, and obtain greater mathematical fluency. f共 Finding the va simplify. substitute a for x and

EXAMPLE 1

»

Evaluate Functions

evaluate. Let f共x兲苷 x ⫺ 1, and c. 3f共b兲 b. f共3b兲 a. f共⫺5 兲 2

d.

Solution

In Example 1, observe that f (3b) 苷 3f (b) and that f (a ⫹ 3) 苷 f (a) ⫹ f (3)

Try Exercise 2, page

Annotated Examples are provided throughout each section and are titled. The annotated steps assist the student in moving from step to step and help explain the solution.

x. • Substitute a ⴙ 3 for • Simplify. substitute 3 • Substitute a for x; for x. • Simplify.

苷 a2 ⫹ 7

»

Annotated Examples

and simplify. • Substitute b for x,

2 2 ⫹ 共3 ⫺ 1兲 f共a兲 ⫹ f共3兲苷共a ⫺ 1兲

e.

f共a兲 ⫹ f共3兲

e.

, and simplify. • Substitute ⴚ5 for x , and simplify. • Substitute 3b for x

2 25 ⫺ 1 苷 24 f共⫺5 兲苷共⫺5兲 ⫺ 1 苷 2 2 苷 9b ⫺ 1 b. f共3b兲苷共3b兲 ⫺ 1 2 2 苷 3b ⫺ 3 c. 3f共b兲苷 3共b ⫺ 1兲 2 ⫺1 d. f共a ⫹ 3兲苷共a ⫹ 3兲苷 a2 ⫹ 6a ⫹ 8

a.

take note

f共a ⫹ 3兲

46

exd by more than one

sente ions are functions repre defined function. Piecewise-defined funct an example of a piecewisen below is ion. The function show 1

In Exercises 1 to 8, evaluate each function. press

1. Given f 共x兲苷 3x ⫺ 1, find a. f 共2兲 d. f

c. f共x兲 f 共0兲苷

b. f 共⫺1兲

冉冊 2 3

冉冊 1 2

Try Exercises

, find up of different 兩x兩 • This function is made 2 x , depending on the pieces, 2x, x , and 4 ⴚ b. f 共⫺2兲 . x value of

References to Try Exercises follow all worked examples, encouraging students to test their understanding by working an exercise similar to the worked example. An icon and a page reference are given below the example, making it easy for the student to navigate to the suggested exercise and back.

on the value of x.

兲, we note that ⫺3 ⬍ ⫺2 For instance, to find f共⫺3 6. ion. Given T共x兲苷 5, find 2x to evaluate the funct

2. Given g共x兲苷 2x 2 ⫹ 3, find

d. g

5. Given f 共x兲苷

x ⬍ ⫺2 2x, ⫺2 ⱕ x ⬍ 1 x 2, a. f 共2兲 4 ⫺ x, x ⱖ 1

ion depends ate thise.funct evalu f. f 共k ⫹ 2兲 to ⫹ ssion used d. f 共2兲 f 共⫺2兲 f 共c 2 ⫹ 4兲 The expression that is and therefore use the expre

e. f 共k兲

a. g共3兲

再

2x. ⴚ2, use the expression x ⬍T共0兲 • Whenb.

b. g共⫺1兲

c. g共0兲

e. g共c兲

instances are5兲some additional f. g共c ⫹ Here d. T共3兲 ⫹ T共1兲

⫺6 f共⫺3 兲苷 2共⫺3 兲 a.苷 T共⫺3兲

ion:

of evaluating this funct e. T共x ⫹ h兲

ⱕ x ⬍ 1, use the • When x satisfies ⴚ2 2 x expression x . 7. Given s共x兲苷 , find 兩x兩• When x ⱖ 1, use the expression 4 ⴚ x.

f共⫺1 兲苷共⫺1兲苷 1 2

3. Given A共w兲苷兹w 2 ⫹ 5, find a. A共0兲

b. A共2兲

c. A共⫺2兲 f共4兲苷 4 ⫺ 4 苷 0 a. s共4兲

d. A共4兲

e. A共r ⫹ 1兲

f. A共⫺c兲

4. Given J共t兲苷 3t ⫺ t, find

d. s共⫺3兲

b. s共5兲

冉冊 1 3

a. J共⫺4兲

b. J共0兲

c. J

d. J共⫺c兲

e. J共x ⫹ 1兲

f. J共x ⫹ h兲

8. Given r共x兲苷 a. r共0兲 d. r

冉冊 1 2

Solutions to Try Exercises

Exercise Set 1.3, page 46

e. s共t兲, t ⬎ 0

2

x , find x⫹4 b. r共⫺1兲 e. r共0.1兲

2. Given g共x兲苷 2x 2 ⫹ 3 a. g共3兲苷 2共3兲 ⫹ 3 苷 18 ⫹ 3 苷 21 2

b. g共⫺1兲苷 2共⫺1兲2 ⫹ 3 苷 2 ⫹ 3 苷 5 c. g共0兲苷 2共0兲2 ⫹ 3 苷 0 ⫹ 3 苷 3

冉冊冉冊

d. g

1 2

1 2

苷2

2

⫹3苷

7 1 ⫹3苷 2 2

e. g共c兲苷 2共c兲 ⫹ 3 苷 2c 2 ⫹ 3 2

f. g共c ⫹ 5兲苷 2共c ⫹ 5兲2 ⫹ 3 苷 2c 2 ⫹ 20c ⫹ 50 ⫹ 3

苷 2c 2 ⫹ 20c ⫹ 53

The complete solution to the Try Exercises can be found in the Solutions to the Try Exercises appendix. This interaction among the examples, the Try Exercises, the Solutions to the Try Exercises, and the exercise sets serves as a checkpoint for students as they read the text, do their homework, and study a section.

For the population/sports team data, r 2 ⬇ 0.87. This means that approximately 87% of the total variation in the dependent variable (number of teams) can be attributed to the state population. This also means that population alone does not predict with certainty the number of sports teams. Other factors, such as climate, are also involved in the number of sports teams. QUESTION

y

What is the coefficient of determination for the odometer reading/ trade-in value data (see page 99), and what is its significance?

Quadratic Regression Models To this point our focus has been linear regression equations. However, there may be a nonlinear relationship between two quantities. The scatter diagram to the left suggests that a quadratic function might be a better model of the data than a linear model. x Nonlinear correlation between x and y.

ANSWER

r 2 ⬇ 0.964. This means that about 96.4% of the total variation in trade-in value can be attributed to the odometer reading.

Question/Answer In each section, the authors pose at least one question that encourages students to pause and think about the concepts presented in the current discussion. To ensure that students do not miss important information, the answer to the question is provided as a footnote on the same page.

xvii

Preface

»

Enhance Conceptual Understanding and Quantitative Reasoning: Using Features for Student Success

y

r

(x, y)

(h, k)

1.2 A Two-Dimensional Coor dinate System and Grap hs 25 circle with center 共h, k兲 and radius r. The point 共x, y兲 is on the circle if is a distance of r units and only if it from the center 共h, k兲. Thus 共x, y兲 is on the circle if and only if 兹共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 苷 r 共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 苷 2 r • Square each side.

x

Figure 1.27

NEW! Immediate Examples of Definitions and Concepts Immediate examples of many definitions enhance understanding.

Standard Form of the Equation of a Circle The standard form of the equation of a circle with center at 共h, k兲 and radiu r is s 共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 苷

Example I

r2

The equation (x ⫺ 2)2 ⫹ (y ⫺ 4)2 苷 32 is in stand ard form, where h 苷 2, k 苷 4, and r 苷 3. The graph of this equa tion is a circle with cente C(2, 4) and radius 3. r

I

The equation (x ⫺ 3)2 ⫹ (y ⫹ 1)2 苷 25 can be written in standard form (x ⫺ 3)2 ⫹ (y ⫺ (⫺1))2 as 苷 52. Note that in the standard form, (x ⫺ h)2 (y ⫺ k)2 are written using and subtraction. Because (y ⫹ 1)2 is written using addition, the expression is rewritten as (y ⫺ (⫺1))2 . The graph of this equation is a circle with center C(3, ⫺1) and radiu s 5. I The equation (x ⫹ 4)2 ⫹ (y ⫹ 2)2 苷 10 can be written in stand (x ⫺ (⫺4))2 ⫹ (y ⫺ (⫺2))2 ard form as 苷 (兹10)2. The graph of this equation is a circle with center C(⫺4, ⫺2) and radius 兹10. If a circle is centered at the origin (0, 0), then h 苷 0 and k 苷 0 and the form of the equation of standard the circle simplifies to x2 ⫹ y2 苷 r 2 • Equation of a circle with center at the origin and radius r.

For instance, x 2 ⫹ y 2 苷 9 is the equation of the circle with center at the radius 兹9 苷 3. origin and QUES TION

Margin Notes Take Notes alert students to a point requiring special attention or are used to amplify the concept under discussion. And Math Matters contain interesting sidelights about mathematics, its history, or its application.

take note f ⫺1共x兲 does not mean f 共x兲苷 2x, f ⫺1共x兲苷

1 . For f 共x兲

1 x but 2

1 1 苷 . f 共x兲 2x

What are the radius and the coordinates of the center of the circle with equation x 2 ⫹ 共 y ⫺ 2兲2 苷 30?

EXAMPLE 6

»

Find the Standard Form of the Equation of a Circle

Find the standard form of the equation of the circle that has center C共⫺4, ⫺2兲 and contains the point P共⫺1, 2兲. Continued

ANSW ER

The radius is 兹30 and

the coordinates of the

䉴

center are 共0, 2兲.

Visualize the Solution For appropriate examples, both algebraic and graphical solutions are provided to help the student visualize the mathematics of the example and to create a link between the algebraic and visual components of a solution.

Updated! To Review Note To Review notes help students recognize the prerequisite skills needed to understand new concepts. These notes direct students to the appropriate page(s) for review, thus decreasing student frustration and creating more opportunities for comprehension. TO REVIEW

Axis of Symmetry See page 000.

This is the standard form of the equation of a parabola with vertex at the origin and the y-axis as its axis of symmetry. The standard form of the equation of a parabola with vertex at the origin and the x-axis as its axis of symmetry is derived in a similar manner.

EXAMPLE 3

ALGEBRAIC SOLUTION

symmetry can be used to verify these statements and provide connections to earlier topics on symmetry.

wk 苷 27

The focus is 共p, 0兲, and the equation of the directrix is x 苷 ⫺p. If p ⬎ 0, the graph of the parabola opens to the right. See Figure 6.4c. If p ⬍ 0, the graph of the parabola opens to the left. See Figure 6.4d.

Im 4 −

0⬚ ⫹ 360⬚k cis for k 苷 0, 1, 2 3

• k ⴝ 0;

0⬚ ⴙ 360⬚(0) ⴝ 0⬚ 3

• k ⴝ 1;

0⬚ ⴙ 360⬚(1) ⴝ 120⬚ 3

苷 3共cos 0⬚ ⫹ i sin 0⬚兲苷3 w1 苷 27 1/3 cis 120⬚

3 3 3 + i 2 2

w2 苷 27 1/3 cis 240⬚

• k ⴝ 2;

0⬚ ⴙ 360⬚(2) ⴝ 240⬚ 3

苷 3共cos 240⬚ ⫹ i sin 240⬚兲 3 3兹3 i 苷⫺ ⫺ 2 2 0⬚ ⫹ 1080⬚ 苷 360⬚. The angles start repeating; thus there are 3 only three cube roots of 27. The three cube roots are graphed in Figure 5.9. For k 苷 3,

»

Try Exercise 28, page 355

3 4 Re

−4

−

苷 3共cos 120⬚ ⫹ i sin 120⬚兲 3 3兹3 苷⫺ ⫹ i 2 2

Axis of Symmetry Is the x-Axis

y 2 苷 4px

1/3

w0 苷 27 1/3 cis 0⬚

The standard form of the equation of a parabola with vertex 共0, 0兲 and the y-axis as its axis of symmetry is x 2 苷 4py The focus is 共0, p兲, and the equation of the directrix is y 苷 ⫺p. If p ⬎ 0, the graph of the parabola opens up. See Figure 6.4a. If p ⬍ 0, the graph of the parabola opens down. See Figure 6.4b. The standard form of the equation of a parabola with vertex 共0, 0兲 and the x-axis as its axis of symmetry is

VISUALIZE THE SOLUTION

Substitute for k to find the three cube roots of 27.

Axis of Symmetry Is the y-Axis

The tests for y-axis and x-axis

Find Cube Roots by De Moivre’s Theorem

Write 27 in trigonometric form: 27 苷 27 cis 0⬚. Then, from De Moivre’s Theorem for finding roots, the cube roots of 27 are

Standard Forms of the Equation of a Parabola with Vertex at the Origin

take note

»

Find the three cube roots of 27.

3 3 3 − i 2 2

−4

Figure 5.9

Note that the arguments of the three cube roots of 27 are 0°, 120°, and 240° and that 兩w0兩苷兩w1兩苷兩w2兩苷 3. In geometric terms, this means that the three cube roots of 27 are equally spaced on a circle centered at the origin with a radius of 3.

xviii

»

Preface

Enhance Conceptual Understanding and Quantitative Reasoning: Using Well-Developed Exercise Sets

Connecting Concepts Each end-of-section exercise set features Connecting Concepts problems which include material from previous sections, are extensions of topics in the section, require data analysis, and offer challenge problems or problems of the form “prove or disprove.”

Connecting Concepts 83. For u 苷具⫺1, 1典, v 苷具2, 3典, and w 苷具5, 5典, find the sum of the three vectors geometrically by using the triangle method of adding vectors. 84. For u 苷具1, 2典, v 苷具3, ⫺2典, and w 苷具⫺1, 4典, find u ⫹ v ⫺ w geometrically by using the triangle method of adding vectors. 85. Find a vector that has initial point 共3, ⫺1兲 and is equivalent to v 苷 2i ⫺ 3j. 86. Find a vector that has initial point 共⫺2, 4兲 and is equivalent to v 苷具⫺1, 3典. 87.

If v 苷 2i ⫺ 5j and w 苷 5i ⫹ 2j have the same initial point, is v perpendicular to w? Why or why

93. Prove that c共v ⭈ w兲苷共cv兲 ⭈ w. 94. Show that the dot product of two nonzero vectors is positive if the angle between the vectors is an acute angle and negative if the angle between the two vectors is an obtuse angle. 95. COMPARISON OF WORK DONE Consider the following two situations. (1) A rope is being used to pull a box up a ramp inclined at an angle a. The rope exerts a force F on the box, and the rope makes an angle u with the ramp. The box is pulled s feet. (2) A rope is being used to pull the same box along a level floor. The rope exerts the same force F on the box. The box is pulled the same s feet. In which case is more work done?

not? 88.

If v 苷具5, 6典 and w 苷具6, 5典 have the same initial point, is v perpendicular to w? Why or why not?

s F

89. Let v 苷具⫺2, 7典. Find a vector perpendicular to v. 90. Let w 苷 4i ⫹ j. Find a vector perpendicular to w.

Revised! Assessing Concepts Assessing Concepts exercises, found at the end of each chapter, have been enhanced with more question types—including fill-inthe-blank, multiple choice, matching, and true/false.

cos a

91. Is the dot product an associative operation? That is, given any nonzero vectors u, v, and w, does

储v储储w储

.

α

Chapter 4 Assessing Con

F

cepts共u ⭈ v兲 ⭈ w 苷 u ⭈ 共v ⭈ w兲?

1. What is an oblique triang

le?

92. Prove that v ⭈ w 苷 w ⭈ v.

2. In triangle ABC, a 4.5, b 6.2, and C 107º. Which law, the Law of Sines or the Law of Cosines, can be used to find c? 3. Which of the follow ing cases, ASA, AAS, SSA, SSS, is known as the ambig

5. Is the dot product of two

θ

(2)

s

6. Let v and w be nonze ro vectors. Is proj v a vector or a w scalar? 7. True or false: The vecto

r

冓

冔

12 5 , is a unit vector. 13 13

8. True or false: i j 0.

or SAS

uous case of the Law

4. In Heron’s formula, what

(1)

θ

of Sines?

does the variable s repres

ent?

vectors a vector or a scalar

?

9. True or false: The Law of Sines can be used to solve any triangle, given two angle s and any side. 10. True or false: If two nonzero vectors are orthog onal, then their dot product is 0.

Ch

Topics for Discussion Each section ends with Topics for Discussion exercises to be used for group discussion or as writing exercises. These exercises are frequently conceptual and focus on key ideas in the section.

Topics for Discussion 1. Discuss the meaning of symmetry of a graph with respect to a line. How do you determine whether a graph has symmetry with respect to the x-axis? with respect to the y-axis? 2. Discuss the meaning of symmetry of a graph with respect to a point. How do you determine whether a graph has symmetry with respect to the origin? 3. What does it mean to reflect a graph across the x-axis or across the y-axis? 4. Explain how the graphs of y › 2x 3 x 2 and y › 2共x兲3 共x兲2 are related. 5. Given the graph of the function y › f 共x兲, explain how to obtain the graph of the function y › f共x 3兲 1. 6. The graph of the step function y › 冀x冁 has steps that are 1 unit wide. Deter1 mine how wide the steps are for the graph of y › x . 3

决冴

Extensive Exercises The authors thoroughly reviewed each exercise set to update applications and to ensure a smooth progression from routine exercises to exercises that are more challenging. The exercises illustrate the many facets of topics discussed in the text. Each exercise set emphasizes skill building, skill maintenance, conceptual understanding, quantitative reasoning, and, as appropriate, applications. Each exercise set is directly proceeded by Topics for Discussion exercises and directly followed by Connecting Concepts exercises and Projects. Each chapter ends with Assessing Concepts exercises, Chapter Review Exercises, and a Chapter Test. Each chapter, except chapter P, includes a Cumulative Review Exercise Set. Answers to all exercises in the Review Exercises, the Chapter Test, and the Cumulative Review Exercises are included in the student answer section. Along with the answer, there is a reference to the section that pertains to each exercise.

Preface

xix

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1 Functions and Graphs 1.1

Equations and Inequalities

1.2

A Two-Dimensional Coordinate System and Graphs

1.3

Introduction to Functions

1.4

Properties of Graphs

1.5

The Algebra of Functions

1.6

Inverse Functions

1.7

Modeling Data Using Regression

The Internet, the World Wide Web, and Modular Functions

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

In 1965, Lawrence Roberts connected two computers, one at MIT and one at UCLA, using a telephone line. This computer connection was the first demonstration of the feasibility of computer networks and led to the eventual establishment of the Internet, first known as ARPANET after the Advanced Research Projects Agency (ARPA), a group founded by the Department of Defense. Originally, ARPANET connected computers at four universities: the University of California, Los Angeles; Stanford University; the University of California, Santa Barbara; and the University of Utah. The first message was sent from UCLA to Stanford in 1969. To expand the Internet, computer programs had to be written that allowed the interaction of many computers. This set of programs, called protocols, was created by Robert Kahn and Vint Cerf. The programs were called TCP/IP (Transmission Control Protocol/Internet Protocol). TCP/IP was universally adopted in 1983 and is still used today to control communication among computers on the Internet. A rough analog to TCP/IP programs is our telephone system. Each telephone has a unique telephone number. Similarly, each computer on the Internet has a unique IP (Internet Protocol) address. TCP programs are similar to telephone services such as call waiting and caller ID. They enable a computer to know when other computers are trying to communicate with it, and what kind of communication is incoming. In 1991, British physicist Tim Berners-Lee developed a program that allowed physicists to exchange information in an efficient manner. His creation was called the World Wide Web. Two functions that can be performed using the Internet and the World Wide Web are credit card purchases and banking transactions. These transactions must be carried out in a secure mode using programs that encrypt credit card numbers and bank account numbers. One method of encryption makes use of a modular function and the inverse of that function. Modular functions are discussed in the Quantitative Reasoning feature on page 114. A basic example of the use of inverse functions to encrypt and decrypt messages is given in Exercise 59 on page 93.

1

2

Chapter 1

Functions and Graphs

Equations and Inequalities

Section 1.1 ■ ■

■

■ ■

■

The Real Numbers Absolute Value and Distance Linear and Quadratic Equations Inequalities Solving Inequalities by the Critical Value Method Absolute Value Inequalities

Math Matters Archimedes (c. 287–212 B.C.) was the first to calculate p with any degree of precision. He was able to show that 3

10 1 p 3 71 7

from which we get the approximation 1 3 ⬇ p. The use of the symbol p 7 for this quantity was introduced by Leonhard Euler (1707–1783) in 1739, approximately 2000 years after Archimedes. A

B

−5 − 4 −3 −2 −1 0

C 1

2

3

4

5

Figure 1.1

The Real Numbers The real numbers are used extensively in mathematics. The set of real numbers is quite comprehensive and contains several unique sets of numbers. The integers are the set of numbers 兵. . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . .其 Recall that the brace symbols, 兵其, are used to identify a set. The positive integers are called natural numbers. a The rational numbers are the set of numbers of the form , where a and b b 3 5 are integers and b 苷 0. Thus the rational numbers include and . Because 4 2 a each integer can be expressed in the form with denominator b 苷 1, the inteb gers are included in the set of rational numbers. Every rational number can be written as either a terminating or a repeating decimal. A number written in decimal form that does not repeat or terminate is called an irrational number. Some examples of irrational numbers are 0.141141114..., 兹2, and p. These numbers cannot be expressed as quotients of integers. The set of real numbers is the union of the sets of rational and irrational numbers. A real number can be represented geometrically on a coordinate axis called a real number line. Each point on this line is associated with a real number called the coordinate of the point. Conversely, each real number can be associated with 7 a point on a real number line. In Figure 1.1, the coordinate of A is , the coor2 dinate of B is 0, and the coordinate of C is 兹2. Given any two real numbers a and b, we say that a is less than b, denoted by a b, if a b is a negative number. Similarly, we say that a is greater than b, denoted by a b, if a b is a positive number. When a equals b, a b is zero. The symbols and are called inequality symbols. Two other inequality symbols, (less than or equal to) and (greater than or equal to), are also used. The inequality symbols can be used to designate sets of real numbers. If a b, the interval notation 共a, b兲 is used to indicate the set of real numbers between a and b. This set of numbers also can be described using set-builder notation: 共a, b兲苷兵x 兩 a x b其

take note The interval notation 关1, 兲 represents all real numbers greater than or equal to 1. The interval notation 共, 4兲 represents all real numbers less than 4.

When reading a set written in set-builder notation, we read 兵x兩 as “the set of x such that.” The expression that follows the vertical bar designates the elements in the set. The set 共a, b兲 is called an open interval. The graph of the open interval consists of all the points on the real number line between a and b, not including a and b. A closed interval, denoted by 关a, b兴, consists of all points between a and b, including a and b. We can also discuss half-open intervals. An example of each type of interval is shown in Figure 1.2.

1.1 − 5 − 4 −3 −2 −1 0

1

2

3

4

5

The open interval (−2, 4)

− 5 − 4 −3 −2 −1 0

1

2

3

4

5

4

5

Equations and Inequalities

共2, 4兲苷兵x 兩 2 x 4其

An open interval

关1, 5兴苷兵x 兩 1 x 5其

A closed interval

关4, 0兲苷兵x 兩 4 x 0其

A half-open interval

共5, 2兴苷兵x 兩 5 x 2其

A half-open interval

3

The closed interval [1, 5]

− 5 − 4 − 3 −2 −1 0

1

2

3

Absolute Value and Distance

The half-open interval [− 4, 0)

− 5 − 4 − 3 −2 −1 0

1

2

3

4

5

The half-open interval (−5, −2]

The absolute value of a real number is a measure of the distance from zero to the point associated with the number on a real number line. Therefore, the absolute value of a real number is always positive or zero. We now give a more formal definition of absolute value.

Figure 1.2

Definition of Absolute Value For a real number a, the absolute value of a, denoted by 兩a兩, is 兩a兩苷

A

a if a 0 a if a 0

B

5 units

− 5 −4 − 3 −2 −1 0

再

1

2

3

4

Figure 1.3

5

The distance d between the points A and B with coordinates 3 and 2, respectively, on a real number line is the absolute value of the difference between the coordinates. See Figure 1.3. d 苷兩2 共3兲兩苷 5 Because the absolute value is used, we could also write d 苷兩共3兲 2兩苷 5 In general, we define the distance between any two points A and B on a real number line as the absolute value of the difference between the coordinates of the points.

5 4

A

3

Definition of the Distance Between Two Points on a Real Number Line

2 1 9

0 −1

d(A, B) =|4 − (−5)|= 9

Let a and b be the coordinates of the points A and B, respectively, on a real number line. Then the distance between A and B, denoted by d共A, B兲, is d共A, B兲苷兩a b兩

−2 −3 −4 −5

B

Figure 1.4

This formula applies to any real number line. It can be used to find the distance between two points on a vertical real number line, as shown in Figure 1.4.

4

Chapter 1

Functions and Graphs

Linear and Quadratic Equations An equation is a statement about the equality of two expressions. Examples of equations follow. 7苷25

x 2 苷 4x 5

3x 2 苷 2共x 1兲 3

The values of the variable that make an equation a true statement are the roots or solutions of the equation. To solve an equation means to find the solutions of the equation. The number 2 is said to satisfy the equation 2x 1 苷 5 because substituting 2 for x produces 2共2兲 1 苷 5, which is a true statement. Definition of a Linear Equation A linear equation in the single variable x is an equation of the form ax b 苷 0, where a 苷 0.

To solve a linear equation in one variable, isolate the variable on one side of the equals sign. EXAMPLE 1

»

Solve a Linear Equation

Solve: 3x 5 苷 2

Solution 3x 5 苷 2 3x 5 5 苷 2 5 3x 苷 7 3x 7 苷 3 3 7 x苷 3 The solution is

• Add 5 to each side of the equation.

• Divide each side of the equation by 3.

7 . 3

»

Try Exercise 6, page 12

An equation may contain more than one variable. For these equations, called literal equations, we may choose to solve for any one of the variables.

EXAMPLE 2

Solve

»

Solve a Literal Equation

by 苷 ax fo r y. cy

1.1

Equations and Inequalities

Solution by 苷 ax cy by 共 c y兲苷 ax共 c y兲 cy by 苷 axc axy by axy 苷 axc y共 b ax兲苷 axc axc y苷 b ax

• Multiply each side of the equation by c y. • Simplify. • Add axy to each side of the equation. • Factor y from the left side of the equation. • Divide each side of the equation by b ax.

»

Try Exercise 20, page 12

Math Matters

Definition of a Quadratic Equation

The term quadratic is derived from the Latin word quadr¯are, which means “to make square.” Because the area of a square that measures x units on each side is x 2, we refer to equations that can be written in the form ax 2 bx c 0 as equations that are quadratic in x.

An equation of the form ax 2 bx c 苷 0, a 苷 0, is a quadratic equation in x.

A quadratic equation can be solved by using the quadratic formula. The Quadratic Formula The solution of the quadratic equation ax 2 bx c 苷 0, a 苷 0, is given by x苷

QUESTION

Math Matters

ax 3 bx 2 cx d 苷 0 and the general quartic 3

»

Solve a Quadratic Equation

Solve by using the quadratic formula: 2x 2 4x 1 苷 0

Solution We have a 苷 2, b 苷 4, and c 苷 1.

ax bx cx dx e 苷 0 4

For 2x 2 3x 1 苷 0, what are the values of a, b, and c?

EXAMPLE 3

There exists a general procedure to solve “by radicals” the general cubic

b 兹b 2 4ac 2a

共4兲兹共4兲2 4共2兲共1兲 4 兹16 8 苷 2共2兲 4 4 兹8 4 2 兹2 2 兹2 苷苷苷 4 4 2

2

x苷

However, it has been proved that there are no general procedures that can be used to solve “by radicals” general equations of degree 5 or larger.

The solutions are

2 兹2 2 兹2 and . 2 2

»

Try Exercise 30, page 12

ANSWER

a 苷 2, b 苷 3, c 苷 1

5

6

Chapter 1

Functions and Graphs Although every quadratic equation can be solved using the quadratic formula, it is sometimes easier to factor and use the zero product principle.

Zero Product Principle If a and b are algebraic expressions, then ab 苷 0 if and only if a 苷 0 or b 苷 0. Example

To solve 2x 2 x 6 苷 0, first factor the polynomial. 2x 2 x 6 苷 0 共2x 3兲共x 2兲苷 0 2x 3 苷 0 or x 2 苷 0 3 x苷 x 苷 2 2 The solutions are

EXAMPLE 4

• Zero product principle

3 and 2. 2

»

Solve by Using the Zero Product Principle

Solve: 共2x 1兲共x 3兲苷 x 2 x 4

Solution 共2x 1兲共x 3兲苷 x 2 x 4 2x 2 7x 3 苷 x 2 x 4 x 2 8x 7 苷 0 共x 7兲共x 1兲苷 0 x 7 苷 0 or x 1 苷 0 x苷7 x苷1

• Expand the binomial product. • Write as ax 2 bx c 0. • Factor. • Apply the zero product principle.

The solutions are 1 and 7.

»

Try Exercise 44, page 12

Inequalities A statement that contains the symbol , , , or is called an inequality. An inequality expresses the relative order of two mathematical expressions. The solution set of an inequality is the set of real numbers each of which, when substituted for the variable, results in a true inequality. The inequality x 4 is true for any

1.1

Equations and Inequalities

7

17 are all solutions of x 4. 3 The solution set of the inequality can be written in set-builder notation as 兵x 兩 x 4其 or in interval notation as 共4, 兲. Equivalent inequalities have the same solution set. We solve an inequality by producing simpler but equivalent inequalities until the solutions are found. To produce these simpler but equivalent inequalities, we apply the following properties. value of x greater than 4. For instance, 5, 兹21, and

Properties of Inequalities Let a, b, and c be real numbers. 1. Addition Property Adding the same real number to each side of an inequality preserves the direction of the inequality symbol. a b and a c b c are equivalent inequalities. 2. Multiplication Properties a. Multiplying each side of an inequality by the same positive real number preserves the direction of the inequality symbol. If c 0, then a b and ac bc are equivalent inequalities. b. Multiplying each side of an inequality by the same negative real number changes the direction of the inequality symbol. If c 0, then a b and ac bc are equivalent inequalities.

Note the difference between Property 2a and Property 2b. Property 2a states that an equivalent inequality is produced when each side of a given inequality is multiplied by the same positive real number and that the direction of the inequality symbol is not changed. By contrast, Property 2b states that when each side of a given inequality is multiplied by a negative real number, we must reverse the direction of the inequality symbol to produce an equivalent inequality. For instance, 2b 6 and b 3 are equivalent inequalities. (We multiplied 1 each side of the first inequality by , and we changed the “less than” symbol 2 to a “greater than” symbol.) Because subtraction is defined in terms of addition, subtracting the same real number from each side of an inequality does not change the direction of the inequality symbol. Because division is defined in terms of multiplication, dividing each side of an inequality by the same positive real number does not change the direction of the inequality symbol, and dividing each side of an inequality by a negative real number changes the direction of the inequality symbol.

8

Chapter 1

Functions and Graphs

take note

EXAMPLE 5

Solutions of inequalities are often

»

Solve an Inequality

Solve 2共x 3兲 4x 10. Write the solution set in set-builder notation.

stated using set-builder notation or interval notation. For instance,

Solution

the real numbers that are

2共x 3兲 4x 10 2x 6 4x 10 2x 4

solutions of the inequality in Example 5 can be written in set notation as 兵x 兩 x 2其 or in interval notation as 共2, 兲.

• Use the distributive property. • Subtract 4x and 6 from each side of the inequality.

x 2

• Divide each side by 2 and reverse the inequality symbol.

The solution set is 兵x 兩 x 2其.

»

Try Exercise 58, page 12

Solving Inequalities by the Critical Value Method Any value of x that causes a polynomial in x to equal zero is called a zero of the polynomial. For example, 4 and 1 are both zeros of the polynomial x 2 3x 4, because 共4兲2 3共4兲 4 苷 0 and 12 3 1 4 苷 0. A Sign Property of Polynomials Nonzero polynomials in x have the property that for any value of x between two consecutive real zeros, either all values of the polynomial are positive or all values of the polynomial are negative. In our work with inequalities that involve polynomials, the real zeros of the polynomial are also referred to as critical values of the inequality, because on a number line they separate the real numbers that make the inequality true from those that make it false. In Example 6 we use critical values and the sign property of polynomials to solve an inequality. EXAMPLE 6

»

Solve a Polynomial Inequality

Solve: x 2 3x 4 0

Solution Factoring the polynomial x 2 3x 4 produces the equivalent inequality 共x 4兲共x 1兲 0 −5 −4 −3 −2 −1

0

1

Figure 1.5

2

3

4

5

Thus the zeros of the polynomial x 2 3x 4 are 4 and 1. They are the critical values of the inequality x 2 3x 4 0. They separate the real number line into the three intervals shown in Figure 1.5. To determine the intervals on which x 2 3x 4 0, pick a number called a test value from each of the three intervals and then determine

1.1

Equations and Inequalities

9

whether x 2 3x 4 0 for each of these test values. For example, in the interval 共, 4兲, pick a test value of, say, 5. Then x 2 3x 4 苷共5兲2 3共5兲 4 苷 6 Because 6 is not less than 0, by the sign property of polynomials, no number in the interval 共, 4兲 makes x 2 3x 4 0. Now pick a test value from the interval 共4, 1兲, say, 0. When x 苷 0, x 2 3x 4 苷 0 2 3共0兲 4 苷 4 Because 4 is less than 0, by the sign property of polynomials, all numbers in the interval 共4, 1兲 make x 2 3x 4 0. If we pick a test value of 2 from the interval 共1, 兲, then x 2 3x 4 苷共2兲2 3共2兲 4 苷 6 Because 6 is not less than 0, by the sign property of polynomials, no number in the interval 共1, 兲 makes x 2 3x 4 0. The following table is a summary of our work. Interval

Test Value x

x 2 3x 4 v? 0

5

共5兲2 3共5兲 4 0 6 0 False

共, 4兲

− 5 − 4 −3 −2 −1

0

1

Figure 1.6

2

3

4

5

共4, 1兲

0

共0兲2 3共0兲 4 0 4 0 True

共1, 兲

2

共2兲2 3共2兲 4 0 6 0 False

In interval notation, the solution set of x 2 3x 4 0 is 共4, 1兲. The solution set is graphed in Figure 1.6. Note that in this case the critical values 4 and 1 are not included in the solution set because they do not make x 2 3x 4 less than 0.

»

Try Exercise 66, page 12

To avoid the arithmetic in Example 6, we often use a sign diagram. For example, note that the factor 共x 4兲 is negative for all x 4 and positive for all x 4. The factor 共x 1兲 is negative for all x 1 and positive for all x 1. These results are shown in Figure 1.7. (x + 4)

−

+

+

+

+

+

+

+

+

(x − 1)

−

−

−

−

−

+

+

+

+

−5 −4 − 3 −2 −1

0

2

3

4

5

1

Figure 1.7

To determine on which interval(s) the product 共x 4兲共x 1兲 is negative, we examine the sign diagram to see where the factors have opposite signs. This occurs only on the interval 共4, 1兲, where 共x 4兲 is positive and 共x 1兲 is negative. Thus the original inequality is true only on the interval 共4, 1兲.

10

Chapter 1

Functions and Graphs Following is a summary of the steps used to solve polynomial inequalities by the critical value method.

Solving a Polynomial Inequality by the Critical Value Method 1. Write the inequality so that one side of the inequality is a nonzero polynomial and the other side is 0. 2. Find the real zeros of the polynomial. They are the critical values of the original inequality. 3. Use test values to determine which of the intervals formed by the critical values are to be included in the solution set. 4. Any critical value that satisfies the original inequality is an element of the solution set.

Absolute Value Inequalities − 5 −4 −3 −2 −1

0

1

2

3

4

5

2

3

4

5

Figure 1.8

− 5 −4 −3 −2 −1

0

1

Figure 1.9

The solution set of the absolute value inequality 兩x 1兩 3 is the set of all real numbers whose distance from 1 is less than 3. Therefore, the solution set consists of all numbers between 2 and 4. See Figure 1.8. In interval notation, the solution set is 共2, 4兲. The solution set of the absolute value inequality 兩x 1兩 3 is the set of all real numbers whose distance from 1 is greater than 3. Therefore, the solution set consists of all real numbers less than 2 or greater than 4. See Figure 1.9. In interval notation, the solution set is 共, 2兲共4, 兲. The following properties are used to solve absolute value inequalities.

Properties of Absolute Value Inequalities For any variable expression E and any nonnegative real number k, 兩E兩 k if and only if k E k 兩E兩 k if and only if E k or E k

EXAMPLE 7

»

Solve an Absolute Value Inequality

Solve: 兩2 3x兩 7

Solution 兩2 3x兩 7 implies 7 2 3x 7. Solve this compound inequality.

1.1

Equations and Inequalities

11

7 2 3x 7 9 3x 5 3

5 − 3 − 4 −3

x

• Subtract 2 from each of the three parts of the inequality.

5 3

• Multiply each part of the inequality by

1 and 3

reverse the inequality symbols.

−2 −1

0

1

2

3

冉冊

4

In interval notation, the solution set is given by

Figure 1.10

5 , 3 . See Figure 1.10. 3

»

Try Exercise 80, page 13

EXAMPLE 8

take note Some inequalities have a solution

»

Solve an Absolute Value Inequality

Solve: 兩4x 3兩 5

set that consists of all real numbers. For example, 兩x 9兩 0

Solution

is true for all values of x. Because

兩4x 3兩 5 implies 4x 3 5 or 4x 3 5. Solving each of these inequalities produces

an absolute value is always nonnegative, the equation is always

4x 3 5 4x 2 1 x 2

true.

− − 4 −3

−2 −1

1 2 0

1

Figure 1.11

2

3

4

冉

Therefore, the solution set is ,

or

4x 3 5 4x 8 x2

1 2

册

关2, 兲. See Figure 1.11.

»

Try Exercise 78, page 12

Topics for Discussion 1. Discuss the similarities and differences among natural numbers, integers, rational numbers, and real numbers. 2. Discuss the differences among an equation, an inequality, and an expression. 3. Is it possible for an equation to have no solution? If not, explain why. If so, give an example of an equation with no solution. 4. Is the statement 兩x兩苷 x ever true? Explain why or why not. 5. How do quadratic equations in one variable differ from linear equations in one variable? Explain how the method used to solve an equation depends on whether it is a linear or a quadratic equation.

12

Chapter 1

Functions and Graphs

Exercise Set 1.1 In Exercises 1 to 18, solve and check each equation.

11. 2x 10 苷 40

12. 3y 20 苷 2

13. 5x 2 苷 2x 10

14. 4x 11 苷 7x 20

37. 兹2 x 2 3x 兹2 苷 0

38. 2x 2 兹5 x 3 苷 0

39. x 2 苷 3x 5

40. x 2 苷 7x 1

15. 2共x 3兲 5 苷 4共x 5兲

In Exercises 41 to 48, solve each quadratic equation by factoring and applying the zero product property.

16. 6共5s 11兲 12共2s 5兲苷 0

41. x 2 2x 15 苷 0

42. y 2 3y 10 苷 0

1 x 5苷 4 2

43. 8y 2 189y 72 苷 0

44. 12w 2 41w 24 苷 0

45. 3x 2 7x 苷 0

46. 5x 2 苷 8x

1 19 1 x7 x苷 2 4 2

47. 共x 5兲2 9 苷 0

48. 共3x 4兲2 16 苷 0

17.

19.

3 1 2 x 苷 4 2 3

18.

2 1 x5苷 x3 3 2

10.

11. 0.2x 0.4 苷 3.6 13.

12. 0.04x 0.2 苷 0.07

3 3 共n 5兲共n 11兲苷 0 5 4

14.

2 5 共 p 11兲共2p 5兲苷 0 7 5

15. 3共x 5兲共x 1兲苷共3x 4兲共x 2兲 16. 5共x 4兲共x 4兲苷共x 3兲共5x 4兲

In Exercises 49 to 58, use the properties of inequalities to solve each inequality. Write answers using interval notation.

49. 2x 3 11

50. 3x 5 16

51. x 4 3x 16

52. 5x 6 2x 1

53. 6x 1 19

54. 5x 2 37

55. 3共x 2兲 5x 7

56. 4共x 5兲 2x 15

57. 4共3x 5兲 2共x 4兲

58. 3共x 7兲 5共2x 8兲

17. 0.08x 0.12共4000 x兲苷 432 18. 0.075y 0.06共10,000 y兲苷 727.50 In Exercises 19 to 26, solve each equation for the indicated variable.

19. x 2y 苷 8; y

20. 3x 5y 苷 15; y

21. 2x 5y 苷 10; x

22. 5x 4y 苷 10; x

23. ay by 苷 c; y

24. ax by 苷 c; y

25. x 苷

y ;y 1y

26. x 苷

2y 3 ;y y1

In Exercises 27 to 40, solve by using the quadratic formula.

27. x 2 2x 15 苷 0

28. x 2 5x 24 苷 0

29. x 2 x 1 苷 0

30. x 2 x 2 苷 0

31. 2x 2 4x 1 苷 0

32. 2x 2 4x 1 苷 0

33. 3x 2 5x 3 苷 0

34. 3x 2 5x 4 苷 0

1 2 3 35. x x1苷0 2 4

2 2 1 36. x 5x 苷 0 3 2

In Exercises 59 to 66, use the critical value method to solve each inequality. Use interval notation to write each solution set.

59. x 2 7x 0

60. x 2 5x 0

61. x 2 7x 10 0

62. x 2 5x 6 0

63. x 2 3x 28

64. x 2 x 30

65. 6x 2 4 5x

66. 12x 2 8x 15

In Exercises 67 to 84, use interval notation to express the solution set of each inequality.

67. 兩x兩 4

68. 兩x兩 2

69. 兩x 1兩 9

70. 兩x 3兩 10

71. 兩x 3兩 30

72. 兩x 4兩 2

73. 兩2x 1兩 4

74. 兩2x 9兩 7

75. 兩x 3兩 5

76. 兩x 10兩 2

77. 兩3x 10兩 14

78. 兩2x 5兩 1

1.1 79. 兩4 5x兩 24

80. 兩3 2x兩 5

81. 兩x 5兩 0

82. 兩x 7兩 0

83. 兩x 4兩 0

84. 兩2x 7兩 0

85. GEOMETRY The perimeter of a rectangle is 27 centimeters, and its area is 35 square centimeters. Find the length and width of the rectangle. 86. GEOMETRY The perimeter of a rectangle is 34 feet and its area is 60 square feet. Find the length and width of the rectangle. 87. RECTANGULAR ENCLOSURE A gardener wishes to use 600 feet of fencing to enclose a rectangular region and subdivide the region into two smaller rectangles. The total enclosed area is 15,000 square feet. Find the dimensions of the enclosed region.

w

l

88. RECTANGULAR ENCLOSURE A farmer wishes to use 400 yards of fencing to enclose a rectangular region and subdivide the region into three smaller rectangles. If the total enclosed area is 400 square yards, find the dimensions of the enclosed region.

Equations and Inequalities

Account Plan

Monthly Fee

Charge per Check

LowCharge

$5.00

$.01

FeeSaver

$1.00

$.08

13

90. PERSONAL FINANCE You can rent a car for the day from company A for $29.00 plus $0.12 a mile. Company B charges $22.00 plus $0.21 a mile. Find the number of miles m (to the nearest mile) per day for which it is cheaper to rent from company A. 91. PERSONAL FINANCE A sales clerk has a choice between two payment plans. Plan A pays $100.00 a week plus $8.00 a sale. Plan B pays $250.00 a week plus $3.50 a sale. How many sales per week must be made for plan A to yield the greater paycheck? 92. PERSONAL FINANCE A video store offers two rental plans. The yearly membership fee and the daily charge per video are shown below. How many videos can be rented per year if the No-fee plan is to be the less expensive of the plans?

THE VIDEO STORE Rental Plan

Yearly Fee

Daily Charge per Video

Low-rate

$15.00

$1.49

No-fee

None

$1.99

w

l

89. PERSONAL FINANCE A bank offers two checking account plans. The monthly fee and charge per check for each plan are shown in the table at the top of the next column. Under what conditions is it less expensive to use the LowCharge plan?

93. AVERAGE TEMPERATURES The average daily minimum-tomaximum temperature range for the city of Palm Springs during the month of September is 68 F to 104 F. What is the corresponding temperature range measured on the Celsius temperature scale? (Hint: Let F be the average daily tem9 perature. Then 68 F 104. Now substitute C 32 for 5 F and solve the resulting inequality for C.)

14

Chapter 1

Functions and Graphs

Connecting Concepts 94. A GOLDEN RECTANGLE The ancient Greeks defined a rectangle as a golden rectangle if its length l and its width w satisfied the equation

99. HEIGHT OF A PROJECTILE The equation s 苷 16t 2 v0 t s0 gives the height s, in feet above ground level, at the time t seconds after an object is thrown directly upward from a height s0 feet above the ground with an initial velocity of v0 feet per second. A ball is thrown directly upward from ground level with an initial velocity of 64 feet per second. Find the time interval during which the ball attains a height of than 48 feet.

w l 苷 w lw a. Solve this formula for w. b. If the length of a golden rectangle is 101 feet, determine its width. Round to the nearest hundredth of a foot. 95. SUM OF NATURAL NUMBERS The sum S of the first n natural numbers 1, 2, 3, . . . , n is given by the formula S苷

n 共n 1兲 2

How many consecutive natural numbers starting with 1 produce a sum of 253?

100. HEIGHT OF A PROJECTILE A ball is thrown directly upward from a height of 32 feet above a stream with an initial velocity of 80 feet per second. Find the time interval during which the ball will be more than 96 feet above the stream. (Hint: See Exercise 99.)

96. NUMBER OF DIAGONALS The number of diagonals D of a polygon with n sides is given by the formula

80 ft/sec

n D 苷共n 3兲 2

32 ft

a. Determine the number of sides of a polygon with 464 diagonals. b. Can a polygon have 12 diagonals? Explain. 97. REVENUE The monthly revenue R for a product is given by R 苷 420x 2x 2, where x is the price in dollars of each unit produced. Find the interval in terms of x for which the monthly revenue is greater than zero.

101. GEOMETRY The length of the side of a square has been measured accurately to within 0.01 foot. This measured length is 4.25 feet. a. Write an absolute value inequality that describes the relationship between the actual length of each side of the square s and its measured length.

98. ABSOLUTE VALUE INEQUALITIES Write an absolute value inequality to represent all real numbers within a. 8 units of 3

b. Solve the absolute value inequality you found in a. for s.

b. k units of j (assume k 0)

Projects 1.

TEACHING MATHEMATICS Prepare a lesson that you could use to explain to someone how to solve linear and quadratic equations. Be sure to include an explanation of the differences between these two types of equations and the different methods that are used to solve them.

2.

CUBIC EQUATIONS Write an essay on the development of the solution of the cubic equation. An excellent source of information is the chapter “Cardano and the Solution of the Cubic” in Journey Through Genius by William Dunham (New York: Wiley, 1990). Another excellent source is A History of Mathematics: An Introduction by Victor J. Katz (New York: Harper Collins, 1993).

1.2

Section 1.2 ■

■

■ ■ ■

Cartesian Coordinate Systems The Distance and Midpoint Formulas Graph of an Equation Intercepts Circles, Their Equations, and Their Graphs

A Two-Dimensional Coordinate System and Graphs

15

A Two-Dimensional Coordinate System and Graphs P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A1.

PS1. Evaluate

x1 x2 when x1 苷 4 and x2 苷 7. 2

PS2. Simplify 兹50. PS3. Is y 苷 3x 2 a true equation when y 苷 5 and x 苷 1? [1.1] PS4. If y 苷 x 2 3x 2, find y when x 苷 3. [1.1] PS5. Evaluate 兩x y兩 when x 苷 3 and y 苷 1. [1.1] PS6. Evaluate 兹b2 4ac when a 苷 2, b 苷 3, and c 苷 2.

Cartesian Coordinate Systems take note Abscissa comes from the same root word as scissors. An open pair of scissors looks like an x.

Math Matters The concepts of analytic geometry developed over an extended period of time, culminating in 1637 with the publication of two works: Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences by Ren´e Descartes (1596–1650) and Introduction to Plane and Solid Loci by Pierre de Fermat. Each of these works was an attempt to integrate the study of geometry with the study of algebra. Of the two mathematicians, Descartes is usually given most of the credit for developing analytic geometry. In fact, Descartes became so famous in La Haye, the city in which he was born, that it was renamed La Haye-Descartes.

Each point on a coordinate axis is associated with a number called its coordinate. Each point on a flat, two-dimensional surface, called a coordinate plane or xy-plane, is associated with an ordered pair of numbers called coordinates of the point. Ordered pairs are denoted by 共a, b兲, where the real number a is the x-coordinate or abscissa and the real number b is the y-coordinate or ordinate. The coordinates of a point are determined by the point’s position relative to a horizontal coordinate axis called the x-axis and a vertical coordinate axis called the y-axis. The axes intersect at the point 共0, 0兲, called the origin. In Figure 1.12, the axes are labeled such that positive numbers appear to the right of the origin on the x-axis and above the origin on the y-axis. The four regions formed by the axes are called quadrants and are numbered counterclockwise. This two-dimensional coordinate system is referred to as a Cartesian coordinate system in honor of René Descartes.

y Quadrant II Horizontal axis −4

4 2

−2 −2

Quadrant III

−4

Quadrant I Vertical axis 2 Origin

4

Quadrant IV

Figure 1.12

x

16

Chapter 1

Functions and Graphs

take note The notation (a, b) was used earlier to denote an interval on a one-

To plot a point P共a, b兲 means to draw a dot at its location in the coordinate plane. In Figure 1.13 we have plotted the points 共4, 3兲, 共3, 1兲, 共2, 3兲, 共3, 2兲, 共0, 1兲, 共1, 3兲, and 共3, 1兲. Note that 共1, 3兲 and 共3, 1兲 are not the same point. The order in which the coordinates of an ordered pair are listed is important.

dimensional number line. In this y

section, (a, b) denotes an ordered pair in a two-dimensional plane.

4

This should not cause confusion in

(1, 3)

(4, 3)

(0, 1)

(3, 1)

2

future sections because as each (−3, 1)

mathematical topic is introduced,

−4

it will be clear whether a one-

−2

2 −2

dimensional or a two-dimensional

4

x

(3, −2)

(−2, −3)

coordinate system is involved.

−4

Figure 1.13

Data often are displayed in visual form as a set of points called a scatter diagram or scatter plot. For instance, the scatter diagram in Figure 1.14 shows the current and projected revenues of Web-filtering software

vendors. (Web-filtering software allows businesses to control which Internet sites are available to employees while at work.) The point whose coordinates are approximately (2005, 520) means that in the year 2005, approximately $520 million in revenues were generated by companies that supplied this software. The line segments that connect the points in Figure 1.14 help illustrate trends.

Revenue from Web-filtering software (in millions of dollars)

R 1000 800 600 400 200 0 2003 2004 2005 2006 2007 2008 2009

t

Figure 1.14 Source: IDC, 2005

QUESTION

From Figure 1.14, will the revenues from Web-filtering software in 2009 be more or less than twice the revenues in 2003?

In some instances, it is important to know when two ordered pairs are equal.

ANSWER

More. The revenue in 2003 was about $350 million. The projected revenue in 2009 is about $925 million, more than twice $350 million.

1.2

17

A Two-Dimensional Coordinate System and Graphs

Definition of the Equality of Ordered Pairs The ordered pairs 共a, b兲 and 共c, d兲 are equal if and only if a 苷 c and b 苷 d. Example

If (3, y) 苷 (x, 2), then x 苷 3 and y 苷 2.

y (1, 2) 2

The Distance and Midpoint Formulas −2

2

4

x

The Cartesian coordinate system makes it possible to combine the concepts of algebra and geometry into a branch of mathematics called analytic geometry. The distance between two points on a horizontal line is the absolute value of the difference between the x-coordinates of the two points. The distance between two points on a vertical line is the absolute value of the difference between the y-coordinates of the two points. For example, as shown in Figure 1.15, the distance d between the points with coordinates 共1, 2兲 and 共1, 3兲 is d 苷兩2 共3兲兩苷 5. If two points are not on a horizontal or vertical line, then a distance formula for the distance between the two points can be developed as follows. The distance between the points P1共x1, y1 兲 and P2共x2, y2 兲 in Figure 1.16 is the length of the hypotenuse of a right triangle whose sides are horizontal and vertical line segments that measure 兩x2 x1兩 and 兩y2 y1兩, respectively. Applying the Pythagorean Theorem to this triangle produces

5 −2 (1, −3)

Figure 1.15

take note B

Pythagorean Theorem Triangle ABC is a c

a

right triangle if and only if

A

b

a 2 b 2 苷 c2.

C

d 2 苷兩x2 x1兩2 兩y2 y1兩2 d 苷兹兩x2 x1兩2 兩y2 y1兩2

y

• Take the square root of each side of the equation. Because d is nonnegative, the negative root is not listed.

P1(x1, y1) y1

苷兹共x2 x1 兲2 共 y2 y1 兲2

d

| y2 – y1| y2

Thus we have established the following theorem.

P2(x 2 , y2) x1

x2

| x2 – x1| Figure 1.16

• Because 兩x2 x1兩2 (x2 x1)2 and 兩y2 y1兩2 (y2 y1)2

x

The Distance Formula The distance d(P1, P2) between the points P1共x1, y1 兲 and P2共x2, y2 兲 is d(P1, P2) 苷兹共x2 x1 兲 2 共y2 y1 兲 2 Example y 8

The distance between P1(3, 4) and P2(7, 2) is given by

6

d(P1, P 2) 苷兹(x2 x1) 2 (y2 y1)2 苷兹[7 (3)] (2 4) 2

2

P1(−3, 4)

d(P1 , P2) = 2 26

2

苷兹10 2 (2)2 苷兹104 苷 2兹26 ⬇ 10.2

4

−4

−2

P2 (7, 2) 2

4

6

8 x

18

Chapter 1 y

Functions and Graphs

P2(x 2 , y2)

The midpoint M of a line segment is the point on the line segment that is equidistant from the endpoints P1共x1, y1 兲 and P2共x2, y2 兲 of the segment. See Figure 1.17.

M(x, y)

P1(x1, y1)

The Midpoint Formula x

The midpoint M of the line segment from P1共x1, y1 兲 to P2共x2, y2 兲 is given by

冉

冊

x1 x2 y1 y2 , 2 2

Figure 1.17

Example

The midpoint of the line segment between P1(2, 6) and P2(3, 4) is given by

冉冉冉冊

冊冊

x1 x2 y1 y2 M苷 , 2 2 苷苷

y 8

P1(−2, 6)

6 4

(2) 3 6 4 , 2 2

M=

1

( 2 , 5)

P2 (3, 4)

2

1 ,5 2

−4

−2

2

4

6 x

The midpoint formula states that the x-coordinate of the midpoint of a line segment is the average of the x-coordinates of the endpoints of the line segment and that the y-coordinate of the midpoint of a line segment is the average of the y-coordinates of the endpoints of the line segment. EXAMPLE 1

»

Find the Midpoint and Length of a Line Segment

Find the midpoint and the length of the line segment connecting the points whose coordinates are P1共4, 3兲 and P2共4, 2兲.

Solution

冉冉冉冊

冊

x1 x2 y1 y2 , 2 2 4 4 3 共2兲苷 , 2 2 1 苷 0, 2

Midpoint 苷

冊

d共P1, P2 兲苷兹共x2 x1 兲2 共 y2 y1 兲2 苷兹共4 共4兲兲2 共2 3兲2 苷兹共8兲2 共5兲2 苷兹64 25 苷兹89

»

Try Exercise 6, page 28

1.2

19

A Two-Dimensional Coordinate System and Graphs

Graph of an Equation The equations below are equations in two variables. y 苷 3x 3 4x 2

Math Matters Maria Agnesi (1718–1799) wrote Foundations of Analysis for the Use of Italian Youth, one of the most successful textbooks of the 18th century. The French Academy authorized a translation into French in 1749, noting that “there is no other book, in any language, which would enable a reader to penetrate as deeply, or as rapidly, into the fundamental concepts of analysis.” A curve that she discusses in her text is given by the equation a3 y苷 2 . x a2 Unfortunately, due to a translation error from Italian to English, the curve became known as the “witch of Agnesi.” y

a

y=

x 2 y 2 苷 25

y苷

x x1

The solution of an equation in two variables is an ordered pair 共x, y兲 whose coordinates satisfy the equation. For instance, the ordered pairs 共3, 4兲, 共4, 3兲, and 共0, 5兲 are some of the solutions of x 2 y 2 苷 25. Generally, there are an infinite number of solutions of an equation in two variables. These solutions can be displayed in a graph. Definition of the Graph of an Equation The graph of an equation in the two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation. Consider y 苷 2x 1. Substituting various values of x into the equation and solving for y produces some of the ordered pairs that satisfy the equation. It is convenient to record the results in a table similar to the one shown below. The graph of the ordered pairs is shown in Figure 1.18. y 2x 1

y

2

2共2兲 1

5

共2, 5兲

1

2共1兲 1

3

共1, 3兲

0

2共0兲 1

1

共0, 1兲

1

2共1兲 1

1

共1, 1兲

2

2共2兲 1

3

共2, 3兲

x

a3 x + a2 2

(x, y)

x

冉

冊冉冊

Choosing some noninteger values of x produces more ordered pairs to graph, 3 5 , 4 , as shown in Figure 1.19. Using still other values such as , 4 and 2 2 of x would result in more and more ordered pairs to graph. The result would be so many dots that the graph would appear as the straight line shown in Figure 1.20, which is the graph of y 苷 2x 1.

−4

y

y

y

4

4

4

2

2

2

−2

2

4

x

−4

−2

2

4

x

−4

−2

2

−2 −4

Figure 1.18

−4

Figure 1.19

−4

Figure 1.20

4

x

20

Chapter 1

Functions and Graphs EXAMPLE 2

»

Draw a Graph by Plotting Points

Graph: x 2 y 苷 1

Solution Solve the equation for y. x 2 y 苷 1 y 苷 x2 1

y (− 2, 5)

(2, 5) 4

(−1, 2)

• Add x 2 to each side.

Select values of x and use the equation to calculate y. Choose enough values of x so that an accurate graph can be drawn. Plot the points and draw a smooth curve through them. See Figure 1.21.

(1, 2) (0, 1)

−4

−2

2

4

x

y 苷 x2 1

Figure 1.21

x

y x2 1

y

(x, y)

2

2

共2兲 1

5

共2, 5兲

1

共1兲 1

2

共1, 2兲

0

共0兲 1

1

共0, 1兲

1

共1兲2 1

2

共1, 2兲

2

共2兲 1

5

共2, 5兲

2

2

2

»

Try Exercise 26, page 28

Integrating Technology Some graphing calculators, such as the TI-83/TI-83 Plus/TI-84 Plus, have a TABLE feature that allows you to create a table similar to the one shown in Example 2. Enter the equation to be graphed, the first value for x, and the increment (the difference between successive values of x). For instance, entering y1 苷 x 2 1, an initial value of x of 2, and an increment of 1 yields a display similar to the one in Figure 1.22. Changing the initial value to 6 and the increment to 2 gives the table in Figure 1.23. Plot1 Plot2 Plot3 \Y 1 = X2+1 \Y2 = TABLE SETUP \Y3 = TblStart=-2 \Y4 = ΔTbl=1 \Y5 = Indpnt: Auto Ask \Y6 = Depend: Auto Ask \Y7 =

X -2 -1 0 1 2 3 4 X=-2

Figure 1.22

Y1 5 2 1 2 5 10 17

1.2

21

A Two-Dimensional Coordinate System and Graphs TABLE SETUP TblStart=-6 ΔTbl=2 Indpnt: Auto Ask Depend: Auto Ask

X -6 -4 -2 0 2 4 6 X=-6

Y1 37 17 5 1 5 17 37

Figure 1.23

With some calculators, you can scroll through the table by using the up- or down-arrow keys. In this way, you can determine many more ordered pairs of the graph.

EXAMPLE 3

»

Graph by Plotting Points

Graph: y 苷兩x 2兩

Solution This equation is already solved for y, so start by choosing an x value and using the equation to determine the corresponding y value. For example, if x 苷 3, then y 苷兩共3兲 2兩苷兩5兩苷 5. Continuing in this manner produces the following table.

y (− 3, 5) 5 (−2, 4) (−1, 3)

(5, 3)

(0, 2)

−2

(1, 1) (2, 0)

y 苷兩x 2兩

Figure 1.24

(4, 2)

When x is

3

2

1

0

1

2

3

4

5

y is

5

4

3

2

1

0

1

2

3

(3, 1) 5

x

Now plot the points listed in the table. Connecting the points forms a V shape, as shown in Figure 1.24.

»

Try Exercise 30, page 28

EXAMPLE 4

Graph: y 2 苷 x

»

Graph by Plotting Points

Solution Solve the equation for y. y2 苷 x y 苷兹x

• Take the square root of each side.

Continued

䉴

22

Chapter 1 y 4 2

Functions and Graphs (16, 4)

(9, 3)

Choose several x values, and use the equation to determine the corresponding y values.

(4, 2) (1, 1)

(0, 0) −2 −4

(1, −1) (4, − 2)

8

12

16 x

When x is

0

1

4

9

16

y is

0

1

2

3

4

(9, −3) (16, − 4)

y 苷x 2

Figure 1.25

Plot the points as shown in Figure 1.25. The graph is a parabola.

»

Try Exercise 32, page 28

Integrating Technology A graphing calculator or computer graphing software can be used to draw the graphs in Examples 3 and 4. These graphing utilities graph a curve in much the same way as you would, by selecting values of x and calculating the corresponding values of y. A curve is then drawn through the points. If you use a graphing utility to graph y 苷兩x 2兩, you will need to use the absolute value function that is built into the utility. The equation you enter will look similar to Y1=abs(X–2). To graph the equation in Example 4, you will enter two equations. The equations you enter will be similar to Y1=兹 (X) Y2=–兹 (X) The graph of the first equation will be the top half of the parabola; the graph of the second equation will be the bottom half.

Intercepts Any point that has an x- or a y-coordinate of zero is called an intercept of the graph of an equation because it is at these points that the graph intersects the x- or the y-axis. Definitions of x-Intercepts and y-Intercepts If 共x1, 0兲 satisfies an equation, then the point 共x1, 0兲 is called an x-intercept of the graph of the equation. If 共0, y1 兲 satisfies an equation, then the point 共0, y1 兲 is called a y-intercept of the graph of the equation.

To find the x-intercepts of the graph of an equation, let y 苷 0 and solve the equation for x. To find the y-intercepts of the graph of an equation, let x 苷 0 and solve the equation for y.

1.2 EXAMPLE 5

»

23

A Two-Dimensional Coordinate System and Graphs

Find x- and y-Intercepts

Find the x- and y-intercepts of the graph of y 苷 x 2 2x 3.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

To find the y-intercept, let x 苷 0 and solve for y.

The graph of y 苷 x 2 2x 3 is shown below. Observe that the graph intersects the x-axis at 共1, 0兲 and 共3, 0兲, the x-intercepts. The graph also intersects the y-axis at 共0, 3兲, the y-intercept.

y 苷 0 2 2共0兲 3 苷 3 To find the x-intercepts, let y 苷 0 and solve for x. 0 苷 x 2 2x 3 0 苷共x 3兲共x 1兲共x 3兲苷 0 or 共x 1兲苷 0 x 苷 3 or x 苷 1

y

Because y 苷 3 when x 苷 0, 共0, 3兲 is a y-intercept. Because x 苷 3 or 1 when y 苷 0, 共3, 0兲 and 共1, 0兲 are x-intercepts. Figure 1.26 confirms that these three points are intercepts.

(4, 5) 4

2 (−1, 0)

(3, 0)

−2

(0, −3)

2

4

x

(2, −3) −4

(1, −4)

y 苷 x 2 2x 3

Figure 1.26

»

Try Exercise 40, page 28

Integrating Technology In Example 5 it was possible to find the x-intercepts by solving a quadratic equation. In some instances, however, solving an equation to find the intercepts may be very difficult. In these cases, a graphing calculator can be used to estimate the x-intercepts. The x-intercepts of the graph of y 苷 x 3 x 4 can be estimated using the ZERO feature of a TI-83/TI-83 Plus/TI-84 Plus calculator. The keystrokes and some sample screens for this procedure are shown on page 24. Continued

䉴

24

Chapter 1

Functions and Graphs

Press Y= . Now enter X^3+X+4. Press ZOOM and select the standard viewing window. Press ENTER .

The “Left Bound?” shown on the bottom of the screen means to move the cursor until it is to the left of an x-intercept. Press ENTER .

10

<

X 3+X+4 ZOOM MEMORY 1 : ZBox 2: Zoom In 3: Zoom Out 4: ZDecimal 5: ZSquare 6: ZStandard 7 ZTrig

<

Y1 = Y2 = Y3 = Y4 = Y5 = Y6 = Y7 =

Press 2nd CALC to access the CALCULATE menu. The y-coordinate of an x-intercept is zero. Therefore, select 2:zero. Press ENTER .

Y1=X 3+X+4

CALCULATE 1 : value 2: zero 3: minimun 4: maximum 5: intersect 6: dy/dx 7: ∫f(x)dx

−10

10

Left bound? X = -1.489362 Y = -.7930613 −10

The “Right Bound?” shown on the bottom of the screen means to move the cursor until it is to the right of the desired x-intercept. Press ENTER .

“Guess?” is shown on the bottom of the screen. Move the cursor until it is approximately on the x-intercept. Press ENTER .

10

10

<

Y1=X 3+X+4

− 10

10

− 10

10

<

<

Y1=X 3+X+4

Right bound? X = -1.06383

The “Zero” shown on the bottom of the screen means that the value of y is 0 when x 苷 1.378797. The x-intercept is about 共1.378797, 0兲.

Y = 1.7321981

− 10

Y1=X 3+X+4

10

Guess? X = -1.276596 Y = .6429403

−10

10

Zero X = -1.378797

−10

Y=0

−10

Circles, Their Equations, and Their Graphs Frequently you will sketch graphs by plotting points. However, some graphs can be sketched merely by recognizing the form of the equation. A circle is an example of a curve whose graph you can sketch after you have inspected its equation.

Definition of a Circle A circle is the set of points in a plane that are a fixed distance from a specified point. The fixed distance is the radius of the circle, and the specified point is the center of the circle.

The standard form of the equation of a circle is derived by using this definition. To derive the standard form, we use the distance formula. Figure 1.27 is a

1.2

25

A Two-Dimensional Coordinate System and Graphs

circle with center 共h, k兲 and radius r. The point 共x, y兲 is on the circle if and only if it is a distance of r units from the center 共h, k兲. Thus 共x, y兲 is on the circle if and only if

y

r

兹共x h兲2 共 y k兲2 苷 r 共x h兲2 共 y k兲2 苷 r 2

(x, y)

(h, k)

• Square each side.

Standard Form of the Equation of a Circle x

Figure 1.27

The standard form of the equation of a circle with center at 共h, k兲 and radius r is 共x h兲2 共 y k兲2 苷 r 2 Example ■

The equation (x 2)2 (y 4)2 苷 32 is in standard form, where h 苷 2, k 苷 4, and r 苷 3. The graph of this equation is a circle with center C(2, 4) and radius 3.

■

The equation (x 3)2 (y 1)2 苷 25 can be written in standard form as (x 3)2 (y (1))2 苷 52. Note that in the standard form, (x h)2 and (y k)2 are written using subtraction. Because (y 1)2 is written using addition, the expression is rewritten as (y (1))2. The graph of this equation is a circle with center C(3, 1) and radius 5.

■

The equation (x 4)2 (y 2)2 苷 10 can be written in standard form as (x (4))2 (y (2))2 苷 (兹10)2. The graph of this equation is a circle with center C(4, 2) and radius 兹10.

If a circle is centered at the origin (0, 0), then h 苷 0 and k 苷 0 and the standard form of the equation of the circle simplifies to x2 y2 苷 r 2

• Equation of a circle with center at the origin and radius r.

For instance, x 2 y 2 苷 9 is the equation of the circle with center at the origin and radius 兹9 苷 3. QUESTION

What are the radius and the coordinates of the center of the circle with equation x 2 共 y 2兲2 苷 30?

EXAMPLE 6

»

Find the Standard Form of the Equation of a Circle

Find the standard form of the equation of the circle that has center C共4, 2兲 and contains the point P共1, 2兲. Continued

ANSWER

The radius is 兹30 and the coordinates of the center are 共0, 2兲.

䉴

26

Chapter 1

Functions and Graphs

y P(−1, 2) −6

Solution 2 x

−2

See the graph of the circle in Figure 1.28. Because the point P is on the circle, the radius r of the circle must equal the distance from C to P. Thus r 苷兹共1 共4兲兲2 共2 共2兲兲2 苷兹9 16 苷兹25 苷 5

C(−4, −2 )

Using the standard form with h 苷 4, k 苷 2, and r 苷 5, we obtain

−6

共x 4兲2 共 y 2兲2 苷 52

Figure 1.28

»

Try Exercise 64, page 28

If we rewrite 共x 4兲2 共 y 2兲2 苷 52 by squaring and combining like terms, we produce x 2 8x 16 y 2 4y 4 苷 25 x 2 y 2 8x 4y 5 苷 0 This form of the equation is known as the general form of the equation of a circle. By completing the square, it is always possible to write the general form x 2 y 2 Ax By C 苷 0 in the standard form 共x h兲2 共 y k兲2 苷 s for some number s. If s 0, the graph is a circle with radius r 苷兹s. If s 苷 0, the graph is the point 共h, k兲. If s 0, the equation has no real solutions and there is no graph.

EXAMPLE 7

»

Find the Center and Radius of a Circle by Completing the Square

Find the center and radius of the circle given by x 2 y 2 6x 4y 3 苷 0

Solution First rearrange and group the terms as shown. 共x 2 6x兲共 y 2 4y兲苷 3

y 2 −2

2

4

6

C(3, −2 )

−6

x 2 y 2 6x 4y 3 苷 0

Figure 1.29

x

Complete the square of 共x 2 6x兲 by adding 9, and complete the square of 共 y 2 4y兲 by adding 4. 共x 2 6x 9兲共 y 2 4y 4兲苷 3 9 4 共x 3兲2 共 y 2兲2 苷 16 共x 3兲2 共 y 共2兲兲2 苷 42

• Add 9 and 4 to each side of the equation.

This equation is the standard form of the equation of a circle and indicates that the graph of the original equation is a circle centered at 共3, 2兲 with radius 4. See Figure 1.29.

»

Try Exercise 66, page 29

1.2

27

A Two-Dimensional Coordinate System and Graphs

Topics for Discussion 1. The distance formula states that the distance d between the points P1共x1, y1 兲 and P2共x2 , y2 兲 is d 苷兹共x2 x1 兲2 共 y2 y1 兲2. Can the distance formula also be written as follows? Explain. d 苷兹共x1 x2 兲2 共 y1 y2 兲2 2. Does the equation 共x 3兲2 共 y 4兲2 苷 6 have a graph that is a circle? Explain. 3. Explain why the graph of 兩x兩兩y兩苷 1 does not contain any points that have a.

a y-coordinate that is greater than 1 or less than 1

b.

an x-coordinate that is greater than 1 or less than 1

4. Discuss the graph of xy 苷 0. 5. Explain how to determine the x- and y-intercepts of a graph defined by an equation (without using the graph).

Exercise Set 1.2 In Exercises 1 and 2, plot the points whose coordinates are given on a Cartesian coordinate system.

1. 共2, 4兲, 共0, 3兲, 共2, 1兲, 共5, 3兲 2. 共3, 5兲, 共4, 3兲, 共0, 2兲, 共2, 0兲 3. PER CAPITA INCOME The following graph, based on data from the Bureau of Economic Analysis, shows the annual per capita income (total income earned divided by population) in the United States for selected years.

34

(5, 32.9)

32

(2, 30.6) (3, 30.8)

30

(6, 33.5)

(4, 31.5)

(1, 29.5) (0, 28.5)

28

0

1

2

3

4

5

6

4. COMPUTER GAMES The graph below shows the results of market research conducted by a developer of computer games. It shows the projected numbers of sales N, in millions, of a game for selected selling prices p in dollars per game. Number sold (in millions)

Per capita income (in thousands of dollars)

I

c. If the percent increase in the per capita income from 2005 to 2006 were the same as the percent increase in the per capita income from 2004 to 2005, what would be the per capita income in 2006?

t

Year (t = 0 corresponds to 1999)

N (8, 80)

80

(15, 70) (22, 60) (27, 50) (31, 40) (34, 30)

60 40 20

0

(36, 20) (37, 10) 5

10

15

20

25

30

35

40

p

Price per game (in dollars)

a. From the graph, what was the per capita income in 2003?

a. Explain the meaning of the ordered pair (22, 60) in the context of this problem.

b. If the increase in the per capita income from 2005 to 2006 were the same as the increase from 2004 to 2005, what would be the per capita income in 2006?

b. Based on the graph, does the projected numbers of sales increase or decrease as the price of this game increases? (Continued)

28

Chapter 1

Functions and Graphs

c. The product of the coordinates of the ordered pairs, R 苷 p N, indicates the revenue R to the company generated by the sale of N games at p dollars per game. Create a scatter diagram of (p, R).

35. y 苷 x 2 2x 8

36. y 苷 x 2 2x 8

37. y 苷 x 2 2

38. y 苷 x 2 1

d. Based on the scatter diagram in c., what happens to the revenue as the price of the game increases?

In Exercises 39 to 48, find the x- and y-intercepts of the graph of each equation. Use the intercepts and additional points as needed to draw the graph of the equation.

In Exercises 5 to 16, find the distance between the points whose coordinates are given.

5. 共6, 4兲, 共8, 11兲

6. 共5, 8兲, 共10, 14兲

7. 共4, 20兲, 共10, 15兲

8. 共40, 32兲, 共36, 20兲

9. 共5, 8兲, 共0, 0兲

10. 共0, 0兲, 共5, 13兲

11. 共兹3, 兹8 兲, 共兹12, 兹27 兲

12. 共兹125, 兹20 兲, 共 6, 2兹5 兲

13. 共a, b兲, 共a, b兲

14. 共a b, b兲, 共a, a b兲

39. 2x 5y 苷 12

40. 3x 4y 苷 15

41. x 苷 y 2 5

42. x 苷 y 2 6

43. x 苷兩y兩 4

44. x 苷 y 3 2

45. x 2 y 2 苷 4

46. x 2 苷 y 2

47. 兩x兩兩y兩苷 4

48. 兩x 4y兩苷 8

In Exercises 49 to 56, determine the center and radius of the circle with the given equation.

15. 共x, 4x兲, 共2x, 3x兲, given that x 0

49. x 2 y 2 苷 36

50. x 2 y 2 苷 49

16. 共x, 4x兲, 共2x, 3x兲, given that x 0

51. 共x 1兲2 共 y 3兲2 苷 49

52. 共x 2兲2 共y 4兲2 苷 25

17. Find all points on the x-axis that are 10 units from the point 共4, 6兲. (Hint: First write the distance formula with 共4, 6兲 as one of the points and 共x, 0兲 as the other point.)

53. 共x 2兲2 共 y 5兲2 苷 25

18. Find all points on the y-axis that are 12 units from the point 共5, 3兲. In Exercises 19 to 24, find the midpoint of the line segment having the given endpoints.

19. 共1, 1兲, 共5, 5兲

20. 共5, 2兲, 共6, 10兲

21. 共6, 3兲, 共6, 11兲

22. 共4, 7兲, 共10, 7兲

23. 共1.75, 2.25兲, 共3.5, 5.57兲

24. 共8.2, 10.1兲, 共2.4, 5.7兲

In Exercises 25 to 38, graph each equation by plotting points that satisfy the equation.

25. x y 苷 4

26. 2x y 苷 1

27. y 苷 0.25x 2

28. 3x 2 2y 苷 4

29. y 苷 2兩x 3兩

30. y 苷兩x 3兩 2

31. y 苷 x 2 3

32. y 苷 x 2 1

33. y 苷

1 共x 1兲2 2

34. y 苷 2共x 2兲2

54. 共x 3兲2 共 y 5兲2 苷 121 55. 共x 8兲2 y 2 苷

1 4

56. x 2 共 y 12兲2 苷 1

In Exercises 57 to 64, find an equation of a circle that satisfies the given conditions. Write your answer in standard form.

57. Center 共4, 1兲, radius r 苷 2 58. Center 共5, 3兲, radius r 苷 4 59. Center

冉冊冉冊

1 1 , , radius r 苷兹5 2 4

60. Center 0,

2 , radius r 苷兹11 3

61. Center 共0, 0兲, passing through 共3, 4兲 62. Center 共0, 0兲, passing through 共5, 12兲 63. Center 共1, 3兲, passing through 共4, 1兲 64. Center 共2, 5兲, passing through 共1, 7兲

1.2

A Two-Dimensional Coordinate System and Graphs

In Exercises 65 to 72, find the center and radius of the graph of the circle. The equations of the circles are written in general form.

65. x 2 y 2 6x 5 苷 0 2

67. x 2 y 2 14x 8y 56 苷 0

75. Find an equation of a circle that has its center at 共7, 11兲 and is tangent to the x-axis. Write your answer in standard form.

68. x y 10x 2y 25 苷 0 2

2

69. 4x 2 4y 2 4x 63 苷 0

76. Find an equation of a circle that has its center at 共2, 3兲 and is tangent to the y-axis. Write your answer in standard form.

70. 9x 9y 6y 17 苷 0 2

2

71. x 2 y 2 x 3y

73. Find an equation of a circle that has a diameter with endpoints 共2, 3兲 and 共4, 11兲. Write your answer in standard form. 74. Find an equation of a circle that has a diameter with endpoints 共7, 2兲 and 共3, 5兲. Write your answer in standard form.

66. x y 6x 4y 12 苷 0 2

29

15 苷0 4

72. x 2 y 2 3x 5y

25 苷0 4

Connecting Concepts In Exercises 77 to 86, graph the set of all points whose x- and y-coordinates satisfy the given conditions. 77. x 苷 1, y 1

78. y 苷 3, x 2

79. y 3

80. x 2

81. xy 0

82. 兩y兩 1,

83. 兩x兩苷 2, 兩y兩苷 3

84. 兩x兩苷 4, 兩y兩苷 1

85. 兩x兩 2, y 2

86. x 1, 兩y兩 3

x 0 y

In Exercises 87 to 90, find the other endpoint of the line segment that has the given endpoint and midpoint. 87. Endpoint 共5, 1兲, midpoint 共9, 3兲 88. Endpoint 共4, 6兲, midpoint 共2, 11兲 89. Endpoint 共3, 8兲, midpoint 共2, 7兲

90. Endpoint 共5, 4兲, midpoint 共0, 0兲 91. Find a formula for the set of all points 共x, y兲 for which the distance from 共x, y兲 to 共3, 4兲 is 5. 92. Find a formula for the set of all points 共x, y兲 for which the distance from 共x, y兲 to 共5, 12兲 is 13. 93. Find a formula for the set of all points 共x, y兲 for which the sum of the distances from 共x, y兲 to 共4, 0兲 and from 共x, y兲 to 共4, 0兲 is 10. 94. Find a formula for the set of all points for which the absolute value of the difference of the distances from 共x, y兲 to 共0, 4兲 and from 共x, y兲 to 共0, 4兲 is 6. 95. Find an equation of a circle that is tangent to both axes, has its center in the second quadrant, and has a radius of 3. 96. Find an equation of a circle that is tangent to both axes, has its center in the third quadrant, and has a diameter of 兹5.

30

Chapter 1

Functions and Graphs

Projects 1. VERIFY A GEOMETRIC THEOREM Use the midpoint formula and the distance formula to prove that the midpoint M of the hypotenuse of a right triangle is equidistant from each of the vertices of the triangle. (Hint: Label the vertices of the triangle as shown in the figure at the right.)

B(0, b)

M

C(0, 0)

A(a, 0)

2. SOLVE A QUADRATIC EQUATION GEOMETRICALLY In the 17th century, Descartes (and others) solved equations by using both algebra and geometry. This project outlines the method Descartes used to solve certain quadratic equations. a. Consider the equation x 2 苷 2ax b 2. Construct a right triangle ABC with AC 苷 a and CB 苷 b. Now draw a circle with center at A and radius a. Let P be the point at which the circle intersects the hypotenuse of the right triangle and Q the point at which an extension of the hypotenuse intersects the circle. Your drawing should be similar to the one at the right. b. Show that QB is a solution of the equation x 2 苷 2ax b 2.

Q T

A

a S

P

c. Show that PB is a solution of the equation x 2 苷 2ax b 2. B

b

C

d. Construct a line parallel to line segment AC and passing through B. Let S and T be the points at which the line intersects the circle. Show that SB and TB are solutions of the equation x 2 苷 2ax b 2.

Section 1.3 ■ ■ ■ ■ ■ ■

■

Relations Functions Function Notation Identifying Functions Graphs of Functions The Greatest Integer Function (Floor Function) Applications

Introduction to Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A2.

PS1. Let y 苷 3x 12. Find x when y 苷 0. [1.1] PS2. Let y 苷 x 2 4x 3. Find x when y 苷 0. [1.1] PS3. Let 共x, y兲 be ordered pairs for y 2 苷 x. Determine values of a and b for the ordered pairs 共9, a兲 and 共9, b兲. [1.2] PS4. Find the length of the line segment connecting P1共4, 1兲 and P2共3, 2兲. [1.2] PS5. What is the greatest integer that is less than 3.2? [1.1] PS6. Suppose y 苷 2x 6. As the values of x increase, do the values of y increase or decrease? [1.2]

1.3

Introduction to Functions

31

Relations

Score

Grade

关90, 100兴

A

关80, 90兲

B

关70, 80兲

C

关60, 70兲

D

关0, 60兲

F

d 苷 16t 2 If t 苷 3, then d 苷 16共3兲2 苷 144 Equation:

The graph in Figure 1.30 defines a correspondence between the length of a pendulum and the time it takes the pendulum to complete one oscillation. For each nonnegative pendulum length, the graph yields only one time. According to the graph, a pendulum length of 2 feet yields an oscillation time of 1.6 seconds, and a length of 4 feet yields an oscillation time of 2.2 seconds, where the time is measured to the nearest tenth of a second. These results can be recorded as the ordered pairs 共2, 1.6兲 and 共4, 2.2兲. Graph: A pendulum's oscillation time y 2.5 Time (in seconds)

Table 1.1

In many situations in science, business, and mathematics, a correspondence exists between two sets. The correspondence is often defined by a table, an equation, or a graph, each of which can be viewed from a mathematical perspective as a set of ordered pairs. In mathematics, any set of ordered pairs is called a relation. Table 1.1 defines a correspondence between a set of percent scores and a set of letter grades. For each score from 0 to 100, there corresponds only one letter grade. The score 94% corresponds to the letter grade of A. Using ordered-pair notation, we record this correspondence as 共94, A兲. The equation d 苷 16t 2 indicates that the distance d that a rock falls (neglecting air resistance) corresponds to the time t that it has been falling. For each nonnegative value t, the equation assigns only one value for the distance d. According to this equation, in 3 seconds a rock will fall 144 feet, which we record as 共3, 144兲. Some of the other ordered pairs determined by d 苷 16t 2 are 共0, 0兲, 共1, 16兲, 共2, 64兲, and 共2.5, 100兲.

(4, 2.2)

2 (2, 1.6)

1.5 1 0.5

1

2

3

4

5

x

Length (in feet)

Figure 1.30

Functions The preceding table, equation, and graph each determines a special type of relation called a function.

32

Chapter 1

Functions and Graphs

Math Matters It is generally agreed among historians that Leonhard Euler (1707–1783) was the first person to use the word function. His definition of function occurs in his book Introduction to Analysis of the Infinite, published in 1748. Euler contributed to many areas of mathematics and was one of the most prolific expositors of mathematics.

Definition of a Function A function is a set of ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinates.

Although every function is a relation, not every relation is a function. For instance, consider 共94, A兲 from the grading correspondence. The first coordinate, 94, is paired with a second coordinate of A. It would not make sense to have 94 paired with A, 共94, A兲, and 94 paired with B, 共94, B兲. The same first coordinate would be paired with two different second coordinates. This would mean that two students with the same score received different grades, one student an A and the other a B! Functions may have ordered pairs with the same second coordinate. For instance, 共94, A兲 and 共95, A兲 are both ordered pairs that belong to the function defined by Table 1.1. A function may have different first coordinates and the same second coordinate. The equation d 苷 16t 2 represents a function because for each value of t there is only one value of d. Not every equation, however, represents a function. For instance, y 2 苷 25 x 2 does not represent a function. The ordered pairs 共3, 4兲 and 共3, 4兲 are both solutions of the equation. However, these ordered pairs do not satisfy the definition of a function: there are two ordered pairs with the same first coordinate but different second coordinates. QUESTION

Does the set 兵共0, 0兲, 共1, 0兲, 共2, 0兲, 共3, 0兲, 共4, 0兲其 define a function?

The domain of a function is the set of all the first coordinates of the ordered pairs. The range of a function is the set of all the second coordinates. In the function determined by the grading correspondence in Table 1.1, the domain is the interval 关0, 100兴. The range is 兵A, B, C, D, F其. In a function, each domain element is paired with one and only one range element. If a function is defined by an equation, the variable that represents elements of the domain is the independent variable. The variable that represents elements of the range is the dependent variable. For the free-fall of a rock situation, we used the equation d 苷 16t 2. The elements of the domain represented the time the rock fell, and the elements of the range represented the distance the rock fell. Thus, in d 苷 16t 2, the independent variable is t and the dependent variable is d. The specific letters used for the independent and dependent variables are not important. For example, y 苷 16x 2 represents the same function as d 苷 16t 2. Traditionally, x is used for the independent variable and y for the dependent variable. Anytime we use the phrase “y is a function of x” or a similar phrase with different letters, the variable that follows “function of” is the independent variable.

ANSWER

Yes. There are no two ordered pairs with the same first coordinate and different second coordinates.

1.3

Introduction to Functions

33

Function Notation Functions can be named by using a letter or a combination of letters, such as f, g, A, log, or tan. If x is an element of the domain of f, then f共x兲, which is read “f of x” or “the value of f at x,” is the element in the range of f that corresponds to the domain element x. The notation “f” and the notation “f共x兲” mean different things. “f” is the name of the function, whereas “f共x兲” is the value of the function at x. Finding the value of f共x兲 is referred to as evaluating f at x. To evaluate f共x兲 at x 苷 a, substitute a for x and simplify.

EXAMPLE 1

»

Evaluate Functions

Let f共x兲苷 x 2 1, and evaluate. f共5兲

a.

b.

f共3b兲

c.

3f共b兲

d.

f共a 3兲

e.

f共a兲 f共3兲

Solution

take note In Example 1, observe that f (3b) 苷 3f (b)

a.

f共5兲苷共5兲2 1 苷 25 1 苷 24

• Substitute 5 for x, and simplify.

b.

f共3b兲苷共3b兲2 1 苷 9b2 1

• Substitute 3b for x, and simplify.

c.

3f共b兲苷 3共b 1兲苷 3b 3

• Substitute b for x, and simplify.

d.

f共a 3兲苷共a 3兲2 1 苷 a2 6a 8

• Substitute a 3 for x.

f共a兲 f共3兲苷共a2 1兲共32 1兲

• Substitute a for x; substitute 3 for x.

2

and that f (a 3) 苷 f (a) f (3)

e.

2

苷 a2 7

• Simplify.

• Simplify.

»

Try Exercise 2, page 46

Piecewise-defined functions are functions represented by more than one expression. The function shown below is an example of a piecewise-defined function.

再

2x, f共x兲苷 x 2, 4 x,

x 2 2 x 1 x1

• This function is made up of different pieces, 2x, x 2, and 4 x , depending on the value of x.

The expression that is used to evaluate this function depends on the value of x. For instance, to find f共3兲, we note that 3 2 and therefore use the expression 2x to evaluate the function. f共3兲苷 2共3兲苷 6

• When x 2, use the expression 2x.

Here are some additional instances of evaluating this function: f共1兲苷共1兲2 苷 1

• When x satisfies 2 x 1, use the expression x 2.

f共4兲苷 4 4 苷 0

• When x 1, use the expression 4 x.

Chapter 1

Functions and Graphs

y 6 4 2 −6 −4 −2

−2

2

4

6

x

The graph of this function is shown in Figure 1.31. Note the use of the open and closed circles at the endpoints of the intervals. These circles are used to show the evaluation of the function at the endpoints of each interval. For instance, because 2 is in the interval 2 x 1, the value of the function at 2 is 4 关 f共2兲苷共2兲2 苷 4兴. Therefore a closed dot is placed at 共2, 4兲. Similarly, when x 苷 1, because 1 is in the interval x 1, the value of the function at 1 is 3 关 f共1兲苷 4 1 苷 3兴.

−4 −6

QUESTION

Evaluate the function f defined at the bottom of page 33 when x 苷 0.5.

Figure 1.31

The absolute value function is another example of a piecewise-defined function. Below is the definition of this function, which is sometimes abbreviated abs共x兲. Its graph (Figure 1.32) is shown at the left.

y 6 4 2 −6 −4 −2

−2

abs共x兲苷 2

4

6

x

Figure 1.32

EXAMPLE 2

»

再

x, x,

x0 x0

Evaluate a Piecewise–Defined Function

Figure 1.33 shows the distance from home plate to the base of the

outfield fence of PETCO Park, the stadium for the San Diego Padres baseball team.

t

y 450

399 f

34

−200 −100

100

200

x

Figure 1.33

The distance T共x兲, in feet, from home plate to the base of the outfield fences can be approximated by the piecewise function

再

兹4.298x 2 2332.11x 412,253, T(x) 苷兹1.000x 2 6.3208x 149,886, 兹2.356x 2 1169.88x 257,818,

141 x 228 104 x 141 236 x 104

where x is the horizontal distance between the vertical axis and a place at the base of the outfield fence. ANSWER

0.5 is in the interval 2 x 1. Therefore, f 共0.5兲苷 0.52 苷 0.25.

1.3

35

Introduction to Functions

For instance, when x 苷 100, the distance to the base of the fence is approximately 399 feet. Use this function to find, to the nearest foot, the distance from home plate to the outfield fence for the following values of x. x 苷 100 feet

a.

b.

x 苷 200 feet

Solution Because 100 is in the interval 104 x 141, evaluate T共x兲苷兹1.000x 2 6.3208x 149,886 when x 苷 100.

a.

T共100兲苷兹1.000共100兲2 6.3208共100兲 149,886 ⬇ 401 The distance to the base of the fence is approximately 401 feet. Because 200 is in the interval 141 x 228, evaluate T共x兲苷兹4.298x 2 2332.11x 412,253 when x 苷 200.

b.

T共200兲苷兹4.298共200 2 兲 2332.11共200兲 412,253 ⬇ 343 The distance to the base of the fence is approximately 343 feet.

»

Try Exercise 10, page 47

Identifying Functions Recall that although every function is a relation, not every relation is a function. In the next example we examine four relations to determine which are functions.

EXAMPLE 3

»

Identify Functions

State whether the relation defines y as a function of x. a.

兵共2, 3兲, 共4, 1兲, 共4, 5兲其

d.

The correspondence between the x values and the y values in Figure 1.34 on page 36.

b.

3x y 苷 1

c.

4x 2 y 2 苷 9

Solution a.

There are two ordered pairs, 共4, 1兲 and 共4, 5兲, with the same first coordinate and different second coordinates. This set does not define y as a function of x.

b.

Solving 3x y 苷 1 for y yields y 苷 3x 1. Because 3x 1 is a unique real number for each x, this equation defines y as a function of x. Continued

䉴

36

Chapter 1

X

Functions and Graphs

Y

c.

Solving 4x 2 y 2 苷 9 for y yields y 苷兹4x 2 9. The right side

兹4x 2 9 produces two values of y for each value of x. For example, when x 苷 0, y 苷 3 or y 苷 3. Thus 4x 2 y 2 苷 9 does not define y as a function of x.

d.

Each x is paired with one and only one y. The correspondence in Figure 1.34 defines y as a function of x.

y1 x1

x2

y3

y2

x3

x4

y4

x5

Figure 1.34

»

Try Exercise 14, page 47

take note You may indicate the domain of a

Sometimes the domain of a function is stated explicitly. For example, each of f, g, and h below is given by an equation followed by a statement that indicates the domain of the function.

function using set notation or

f共x兲苷 x 2, x 0

interval notation. For instance, the

g共t兲苷

domain of f 共x兲苷兹x 3 may be given in each of the following ways:

Set notation: 兵x 兩 x 3其 Interval notation: 关3, 兲

1 ,0t5 t2 4

h共x兲苷 x 2, x 苷 1, 2, 3

Although f and h have the same equation, they are different functions because they have different domains. If the domain of a function is not explicitly stated, then its domain is determined by the following convention.

Domain of a Function Unless otherwise stated, the domain of a function is the set of all real numbers for which the function makes sense and yields real numbers.

EXAMPLE 4

»

Determine the Domain of a Function

Determine the domain of each function. 1 b. f共x兲苷兹x 1 t4 A共s兲苷 s2, where A共s兲 is the area of a square whose sides are s units.

G共t兲苷

a. c.

Solution a.

The number 4 is not an element of the domain because G is undefined when the denominator t 4 equals 0. The domain of G is all real numbers except 4. In interval notation the domain is 共, 4兲共4, 兲.

b.

The radical 兹x 1 is a real number only when x 1 0 or when x 1. Thus, in set notation, the domain of f is 兵x 兩 x 1其.

c.

Because s represents the length of a side of a square, s must be positive. In interval notation the domain of A is 共0, 兲.

»

Try Exercise 28, page 47

1.3

37

Introduction to Functions

Graphs of Functions If a is an element of the domain of a function f, then 共a, f共a兲兲 is an ordered pair that belongs to the function. Definition of the Graph of a Function The graph of a function is the graph of all the ordered pairs that belong to the function.

EXAMPLE 5

»

Graph a Function by Plotting Points

Graph each function. State the domain and the range of each function. f共x兲苷兩x 1兩

a.

再

if x 1 if x 1

2, x,

The domain of f is the set of all real numbers. Write the function as y 苷兩x 1兩. Evaluate the function for several domain values. We have used x 苷 3, 2, 1, 0, 1, 2, 3, and 4.

a.

4

(− 2, 3)

(4, 3)

(− 1, 2)

(3, 2)

(0, 1) −4

n共x兲苷

Solution

y (− 3, 4)

b.

(2, 1) (1, 0)

4

x

x

y 兩x 1兩

3

2

1

0

1

2

3

4

4

3

2

1

0

1

2

3

f共x兲苷兩x 1兩

Figure 1.35

Plot the points determined by the ordered pairs. Connect the points to form the graph in Figure 1.35. Because 兩x 1兩 0, we can conclude that the graph of f extends from a height of 0 upward, so the range is 兵 y 兩 y 0其.

Integrating Technology A graphing utility can be used to draw the graph of a function. For instance, to graph f共x兲苷 x 2 1, enter the equation Y1=x2–1. The graph is shown in Figure 1.37.

The domain is the union of the inequalities x 1 and x 1. Thus the domain of n is the set of all real numbers. For x 1, graph n共x兲苷 2. This results in the horizontal ray in Figure 1.36. The solid circle indicates that the point 共1, 2兲 is part of the graph. For x 1, graph n共x兲苷 x. This produces the second ray in Figure 1.36. The open circle indicates that the point 共1, 1兲 is not part of the graph. Examination of the graph shows that it includes only points whose y values are greater than 1. Thus the range of n is 兵 y 兩 y 1其.

b.

6

−4

4 −2

Figure 1.37

»

Try Exercise 40, page 47

y 4

−4

−2

2

n共x兲苷

再

2, x,

if x 1 if x 1

Figure 1.36

4

x

38

Chapter 1 y

f(x) =

Functions and Graphs 2x + 1 x−1

4 2 −4

−2

x=1 2

4

2x 1 is shown in Figure 1.38. The domain of f is all real x1 numbers except 1. Because f 共1兲 is undefined, there is no point on the graph when x 苷 1. However, there are points on the graph for values of x close to 1. The table below shows some values of the function when x is less than 1 but close to 1. The graph of f 共x兲苷

x

−2

x

−4

f (x)

0.9

0.95

0.99

0.995

0.999

0.9999

28

58

298

598

2998

29,998

Figure 1.38

take note We can also write, “As x l 1,

It appears from the graph and the table that as x gets closer and closer to 1, f 共x兲 becomes smaller and smaller. That is, as x approaches 1 using values of x that are less than 1, f 共x兲 approaches . The notation f 共x兲 l as x l 1

f 共x兲 l .” This is read, “As x approaches 1 from the left, f 共x兲 approaches negative infinity.”

is used to describe this situation. This notation is read “f 共x兲 approaches negative infinity as x approaches 1 from the left.” The negative superscript tells us to use values of x that are less than 1—that is, to the left of 1 on the x-axis. Next we focus on values of x that are close to 1 but greater than 1.

x

1.1

1.05

1.01

1.005

1.001

1.0001

f (x)

32

62

302

602

3002

30,002

This time, from Figure 1.38 and the table, it appears that as x gets closer and closer to 1, f 共x兲 becomes larger and larger. That is, as x approaches 1 using values of x that are greater than 1, f 共x兲 approaches . The notation f 共x兲 l as x l 1 is used to describe this situation. This notation is read “f 共x兲 approaches infinity as x approaches 1 from the right.” The positive superscript tells us to use values of x that are greater than 1—that is, to the right of 1 on the x-axis. 2x 1 The preceding analysis shows that the graph of f共x兲苷 approaches, but x1 does not cross, the vertical line x 苷 1. The dashed line in Figure 1.38 is called a vertical asymptote of the graph.

Definition of Vertical Asymptote

take note Generally, we show the asymptotes of the graph of a function by using dashed lines.

The line x 苷 a is a vertical asymptote of the graph of a function f provided f共x兲 l

or f共x兲 l

as x approaches a from either the left, x l a, or from the right, x l a.

1.3

39

Introduction to Functions

5x . The domain of r is all real 共x 2兲2 numbers except 2. The table below shows the behavior of r共x兲 as x l 2. As an example of this definition, let r共x兲苷

x r 共x兲

1.9

1.95

1.99

1.995

950

3900

99,500

399,000

From the table, it appears that r共x兲 l as x l 2. The following table shows the behavior of r共x兲 as x l 2. y 8

r(x) =

5x (x − 2)2

x r 共x兲

2.1

2.05

2.01

2.005

1050

4100

100,500

401,000

6

From the table, it appears that r共x兲 l as x l 2.

4

Because r共x兲 l

2 −4

−2

4 x=2

Figure 1.39

6 x

as x l 2 and as x l 2, the line x 苷 2 is a vertical

5x , along with the vertical asymptote, is 共x 2兲2 shown in Figure 1.39. In this case, the value of the function is approaching positive infinity on both sides of the vertical asymptote. asymptote. The graph of r共x兲苷

The definition of a function as a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates implies that any vertical line intersects the graph of a function at no more than one point. This is known as the vertical line test. The Vertical Line Test for Functions A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

EXAMPLE 6

»

Apply the Vertical Line Test

State whether the graph is the graph of a function. a.

b.

y

x

y

x

Continued

䉴

40

Chapter 1

Functions and Graphs Solution a.

This graph is not the graph of a function because some vertical lines intersect the graph in more than one point.

b.

This graph is the graph of a function because every vertical line intersects the graph in at most one point.

»

Try Exercise 56, page 48

y

2

−2

2

4

x

Consider the graph in Figure 1.40. As a point on the graph moves from left to right, this graph falls for values of x 2, remains the same height from x 苷 2 to x 苷 2, and rises for x 2. The function represented by the graph is said to be decreasing on the interval 共, 2兴, constant on the interval 关2, 2兴, and increasing on the interval 关2, 兲.

Figure 1.40

Definition of Increasing, Decreasing, and Constant Functions If a and b are elements of an interval I that is a subset of the domain of a function f, then ■

f is increasing on I if f共a兲 f共b兲 whenever a b.

■

f is decreasing on I if f共a兲 f共b兲 whenever a b.

■

f is constant on I if f共a兲苷 f共b兲 for all a and b.

Recall that a function is a relation in which no two ordered pairs that have the same first coordinate have different second coordinates. This means that given any x, there is only one y that can be paired with that x. A one-to-one function satisfies the additional condition that given any y, there is only one x that can be paired with that given y. In a manner similar to applying the vertical line test, we can apply a horizontal line test to identify one-to-one functions.

Horizontal Line Test for a One-To-One Function If every horizontal line intersects the graph of a function at most once, then the graph is the graph of a one-to-one function.

For example, some horizontal lines intersect the graph in Figure 1.41 at more than one point. It is not the graph of a one-to-one function. Every horizontal line intersects the graph in Figure 1.42 at most once. This is the graph of a one-toone function.

1.3

41

Introduction to Functions y

y

x

x

Figure 1.41

Figure 1.42

Some horizontal lines intersect this graph at more than one point. It is not the graph of a one-to-one function.

Every horizontal line intersects this graph at most once. It is the graph of a one-to-one function.

The Greatest Integer Function (Floor Function) take note The greatest integer function is an important function that is often used in advanced mathematics and computer science.

The graphs of some functions do not have any breaks or gaps. These functions, whose graphs can be drawn without lifting the pencil off the paper, are called continuous functions. The graphs of other functions do have breaks or discontinuities. One such function is the greatest integer function or floor function. This function is denoted by various symbols such as 冀x冁, ⎣x⎦ , and int共x兲. The value of the greatest integer function at x is the greatest integer that is less than or equal to x. For instance, ⎣1.1⎦ 苷 2

Integrating Technology Many graphing calculators use the notation int共x兲 for the greatest integer function. Here are screens from a TI-83 Plus/TI-84 Plus. Math NUM CPX PRB 1 : abs( 2: round( 3: iPart( 4: fPart( 5: int( 6: min( int(π) 7↓max( int(-1.1)

QUESTION

冉冊

冀3冁苷 3

int

冉冊

Evaluate. a. int

3 2

5 2

苷2

⎣5⎦ 苷 5

冀p冁苷 3

b. ⎣2⎦

To graph the floor function, first observe that the value of the floor function is constant between any two consecutive integers. For instance, between the integers 1 and 2, we have int共1.1兲苷 1 int共1.35兲苷 1 Between 3 and 2, we have

int共1.872兲苷 1

int共1.999兲苷 1

int共2.98兲苷 3 int共2.4兲苷 3 int共2.35兲苷 3 int共2.01兲苷 3 Using this property of the floor function, we can create a table of values and then graph the floor function. See Figure 1.43 on page 42. 3 -2

int(5/2) 2

ANSWER

冉冊

3 3 , int 2 2 b. Because 2 is the greatest integer less than or equal to 2, ⎣2⎦ 苷 2. a. Because 2 is the greatest integer less than or equal to

苷 2.

42

Chapter 1

Functions and Graphs x

y int(x)

5 x 4

5

4 x 3

4

3 x 2

3

y

4 2

2 x 1

2

1 x 0

1

0x1

0

−2

1x2

1

−4

2x3

2

3x4

3

4x5

4

−4

−2

2

4

x

y = int(x)

Figure 1.43

The graph of the floor function has discontinuities (breaks) whenever x is an integer. The domain of the floor function is the set of real numbers; the range is the set of integers. Because the graph appears to be a series of steps, sometimes the floor function is referred to as a step function.

Integrating Technology Many graphing calculators use the notation int共x兲 for the floor function. The screens below are from a TI-83 Plus/TI-84 Plus graphing calculator. The graph in Figure 1.44 was drawn in “connected” mode. This graph does not show the discontinuities that occur whenever x is an integer. The graph in Figure 1.45 was constructed by graphing the floor function in “dot” mode. In this case the discontinuities at the integers are apparent. 5

5

−5

5

−5

5

−5

−5

y = int(x) connected mode

y = int(x) dot mode

Figure 1.44

Figure 1.45

One application of the floor function is rounding numbers. For instance, suppose a credit card company charges 1.5% interest on an unpaid monthly balance of $237.84. Then the interest charge I is I 苷 237.84 0.015 苷 3.5676

1.3

43

Introduction to Functions

Thus the interest charge is $3.57. Note that the result was rounded to the nearest cent, or hundredth. The computer program that determines the interest charge uses the floor function to calculate the rounding. To round a number N to the nearest kth decimal place, we use the following formula.

take note Observe the effect of adding 0.5. In this case, it increases the units digit that will occupy the

N to the nearest kth decimal 苷

int[10 k(N) 0.5] 10 k

Here is the calculation for the interest owed. In this case, N 苷 237.84 0.015 and k 苷 2 (round to the second decimal place). int[10 2(237.84 0.015) 0.5] 10 2 int[100(3.5676) 0.5] 苷 100 int[356.76 0.5] 苷 100 int[357.26] 苷 100 357 苷苷 3.57 100

hundredths place.

I苷

Now suppose that the monthly balance was $237.57. Then 100(237.57 0.015) 0.5 is 356.355 0.5 苷 356.855. In this case, the units digit is not changed and the final interest charge is $3.56.

Example 7 gives another application of the floor function. EXAMPLE 7

»

Use the Greatest Integer Function to Model Expenses

The cost of parking in a garage is $3 for the first hour or any part of the hour and $2 for each additional hour or any part of an hour thereafter. If x is the time in hours that you park your car, then the cost is given by C共x兲苷 3 2 int共1 x兲, a.

Evaluate C共2兲 and C共2.5兲.

b.

x0

Graph y 苷 C共x兲 for 0 x 5.

Solution a.

C共2兲苷 3 2 int共1 2兲苷 3 2 int共1兲苷 3 2共1兲苷 $5

C共2.5兲苷 3 2 int共1 2.5兲苷 3 2 int共1.5兲苷 3 2共2兲苷 $7

b.

To graph C共x兲 for 0 x 5, consider the value of int共1 x兲 for each of the intervals 0 x 1, 1 x 2, 2 x 3, 3 x 4, and 4 x 5. For instance, when 0 x 1, 0 1 x 1. Thus int共1 x兲苷 0 when 0 x 1. Now consider 1 x 2. When 1 x 2, 1 1 x 0. Thus int共1 x兲苷 1 when 1 x 2. Applying the same reasoning to each of the other intervals gives the following table of values and the graph of C shown in Figure 1.46. Continued

䉴

Chapter 1

Functions and Graphs x

C(x) 3 2 int(1 x)

C(x)

0x1

3

10

1x2

5

2x3

7

3x4

9

4x5

11

Parking cost (in dollars)

44

8 6 4 2

1

2

3

4

5

x

Time (in hours)

C(x) = 3 − 2 int(1 − x)

Figure 1.46

Because C共1兲苷 3, C共2兲苷 5, C共3兲苷 7, C共4兲苷 9, and C共5兲苷 11, we can use a solid circle at the right endpoint of each “step” and an open circle at each left endpoint.

»

Try Exercise 54, page 48

Integrating Technology The function graphed in Example 7 is an example of a function for which a graphing calculator may not produce a graph that is a good representation of the function. You may be required to make adjustments in the MODE, SET UP, or WINDOW of the graphing calculator so that it will produce a better representation of the function. A graph may also require some fine tuning, such as open or solid circles at particular points, to accurately represent the function.

Applications EXAMPLE 8

»

Solve an Application

A car was purchased for $16,500. Assuming the car depreciates at a constant rate of $2200 per year (straight-line depreciation) for the first 7 years, write the value v of the car as a function of time, and calculate the value of the car 3 years after purchase.

Solution Let t represent the number of years that have passed since the car was purchased. Then 2200t is the amount by which the value of the car has depreciated after t years. The value of the car at time t is given by v共t兲苷 16,500 2200t,

0t7

1.3

Introduction to Functions

45

When t 苷 3, the value of the car is v共3兲苷 16,500 2200共3兲苷 16,500 6600 苷 $9900

»

Try Exercise 72, page 49

Often in applied mathematics, formulas are used to determine the functional relationship that exists between two variables.

EXAMPLE 9

»

Solve an Application

A lighthouse is 2 miles south of a port. A ship leaves port and sails east at a rate of 7 mph. Express the distance d between the ship and the lighthouse as a function of time, given that the ship has been sailing for t hours.

Solution Draw a diagram and label it as shown in Figure 1.47. Note that because distance 苷 (rate)(time) and the rate is 7, in t hours the ship has sailed a distance of 7t.

7t

Port 2 mi

关d共t兲兴 2 苷共7t兲2 22 关d共t兲兴 2 苷 49t 2 4 d共t兲苷兹49t 2 4

N

d

Lighthouse

• The Pythagorean Theorem • The sign is not used because the distance d (t ) must be nonnegative.

»

Figure 1.47

Try Exercise 78, page 50

EXAMPLE 10

»

Solve an Application

An open box is to be made from a square piece of cardboard that measures 40 inches on each side. To construct the box, squares that measure x inches on each side are cut from each corner of the cardboard as shown in

x

Figure 1.48. 40 − 2x

a.

Express the volume V of the box as a function of x.

b.

Determine the domain of V.

x x

x

40 − 2 x 40 in.

Solution The length l of the box is 40 2x. The width w is also 40 2x. The height of the box is x. The volume V of a box is the product of its length, its width, and its height. Thus

a.

40 − 2 x

V 苷共40 2x兲2x

x 40 − 2x

The squares that are cut from each corner require x to be larger than 0 inches but less than 20 inches. Thus the domain is 兵x 兩 0 x 20其.

b.

Figure 1.48

»

Try Exercise 74, page 49

46

Chapter 1

Functions and Graphs

Topics for Discussion 1. Discuss the definition of function. Give some examples of relationships that are functions and some that are not functions. 2. What is the difference between the domain and range of a function? 3. How many y-intercepts can a function have? How many x-intercepts can a function have? 4. Discuss how the vertical line test is used to determine whether or not a graph is the graph of a function. Explain why the vertical line test works. 5. What is the domain of f共x兲苷 6. Is 2 in the range of g共x兲苷

兹1 x ? Explain. x2 9

6x 5 ? Explain the process you used to make 3x 1

your decision. 7. Suppose that f is a function and that f共a兲苷 f共b兲. Does this imply that a 苷 b? Explain your answer.

Exercise Set 1.3 In Exercises 1 to 8, evaluate each function.

5. Given f 共x兲苷

1. Given f 共x兲苷 3x 1, find a. f 共2兲 d. f

冉冊 2 3

冉冊

d. g

1 2

冉冊 3 5

b. f 共1兲

c. f 共0兲

a. f 共2兲

b. f 共2兲

c. f

e. f 共k兲

f. f 共k 2兲

d. f 共2兲 f 共2兲

e. f 共c 2 4兲

f. f 共2 h兲

6. Given T共x兲苷 5, find

2. Given g共x兲苷 2x 2 3, find a. g共3兲

1 , find 兩x兩

冉冊 2 7

b. g共1兲

c. g共0兲

a. T共3兲

b. T共0兲

c. T

e. g共c兲

f. g共c 5兲

d. T共3兲 T共1兲

e. T共x h兲

f. T共3k 5兲

a. s共4兲

b. s共5兲

c. s共2兲

d. s共3兲

e. s共t兲, t 0

f. s共t兲, t 0

3. Given A共w兲苷兹w 2 5, find

7. Given s共x兲苷

a. A共0兲

b. A共2兲

c. A共2兲

d. A共4兲

e. A共r 1兲

f. A共c兲

4. Given J共t兲苷 3t 2 t, find

冉冊

a. J共4兲

b. J共0兲

1 c. J 3

d. J共c兲

e. J共x 1兲

f. J共x h兲

8. Given r共x兲苷 a. r共0兲

冉冊

d. r

1 2

x , find 兩x兩

x , find x4 b. r共1兲

c. r共3兲

e. r共0.1兲

f. r共10,000兲

1.3 In Exercises 9 and 10, evaluate each piecewise-defined function for the indicated values.

9. P共x兲苷

再

if x 2 if x 2

3x 1, x 2 11,

c. P共c兲,

再

c2

4, 10. t 9, Q共t兲苷䉴兹t 7,

34. f 共x兲苷兹4 x

35. f 共x兲苷兹4 x 2

36. f 共x兲苷兹12 x 2

1

38. f 共x兲苷

兹x 4

47

1 兹5 x

d. P共k 1兲, k 1 if 0 t 5 if 5 t 8 if 8 t 11

a. Q共0兲 c. Q共n兲,

33. f 共x兲苷兹7 x

37. f 共x兲苷

b. P共兹5 兲

a. P共4兲

Introduction to Functions

b. Q共a兲, 1n2

In Exercises 39 to 46, graph each function. Insert solid circles or open circles where necessary to indicate the true nature of the function.

6a7

d. Q共m2 7兲,

1m2

In Exercises 11 to 20, state whether the equation defines y as a function of x.

11. 2x 3y 苷 7

12. 5x y 苷 8

13. x y 2 苷 2

14. x 2 2y 苷 2

15. y 苷 4 兹x

16. x 2 y 2 苷 9

3

17. y 苷兹 x

18. y 苷兩x兩 5

19. y 2 苷 x 2

20. y 3 苷 x 3

In Exercises 21 to 26, state whether the set of ordered pairs (x, y) defines y as a function of x.

39. f 共x兲苷

再

兩x兩, 2,

再再

if x 1 if x 1

4, 40. g共x兲苷 x 2 4, x,

if x 0 if 0 x 1 if x 1

4, 41. J共x兲苷 x 2, x 5,

if x 1 if 1 x 1 if x 1

1, if x 2 42. K共x兲苷 x 2 4, if 2 x 2 1 x, if x 2 2

冦

43. L共x兲苷

决 31 x冴

for 6 x 6

21. 兵共2, 3兲, 共5, 1兲, 共4, 3兲, 共7, 11兲其

44. M共x兲苷冀x冁 2 for 0 x 4

22. 兵共5, 10兲, 共3, 2兲, 共4, 7兲, 共5, 8兲其

45. N共x兲苷 int共x兲 for 3 x 3

23. 兵共4, 4兲, 共6, 1兲, 共5, 3兲其

46. P共x兲苷 int共x兲 x for 0 x 4

24. 兵共2, 2兲, 共3, 3兲, 共7, 7兲其 25. 兵共1, 0兲, 共2, 0兲, 共3, 0兲其 26.

再冉

冊冉

冊冉冊冎

1 1 1 1 , , , , 3 4 4 3

1 2 , 4 3

In Exercises 27 to 38, determine the domain of the function represented by the given equation.

27. f 共x兲苷 3x 4

28. f 共x兲苷 2x 1

29. f 共x兲苷 x 2 2

30. f 共x兲苷 3x 2 1

4 31. f 共x兲苷 x2

6 32. f 共x兲苷 x5

In Exercises 47 to 52, use the floor function to write and then evaluate an expression that can be used to round the given number to the given place value.

47. N 苷 2.3458; hundredths 48. N 苷 34.567; tenths 49. N 苷 34.05622; thousandths 50. N 苷 109.83; whole number 51. N 苷 0.08951; ten-thousandths 52. N 苷 2.98245; thousandths

48

Chapter 1

Functions and Graphs

53. FIRST-CLASS MAIL In 2006, the cost to mail a first-class letter was given by

56. Use the vertical line test to determine which of the following graphs are graphs of functions.

w0

C共w兲苷 0.39 0.34 int共1 w兲,

a.

b.

y

y 2

2

where C is in dollars and w is the weight of the letter in ounces. −4

a. What was the cost (in 2006) to mail a letter that weighed 2.8 ounces?

−2

3 x

−3

4 x

2

−2

b. Graph C for 0 w 5.

c.

d.

y

y

3

54. INCOME TAX The amount of federal income tax T共x兲 a person owed in 2006 is given by

2

⎧ 0.10x, 0 x 7550 ⎪ 0.15共x 7550兲 755, 7550 x 30,650 ⎪ 0.25共x 30,650兲 4220, 30,650 x 74,200 T共x兲苷 ⎨ 0.28共x 74,200兲 15,107.50, 74,200 x 154,800 ⎪ ⎪ 0.33共x 154,800兲 37,675.50, 154,800 x 336,550 ⎩ 0.35共x 336,550兲 97,653, x 336,550 where x is the adjusted gross income tax of the taxpayer.

−3

3

x

−2

x

2 −2

−3

a. What is the domain of this function?

In Exercises 57 to 66, use the indicated graph to identify the intervals over which the function is increasing, constant, or decreasing.

b. Find the income tax owed by a taxpayer whose adjusted gross income was $31,250.

57.

b.

4 g 2

2 f

55. Use the vertical line test to determine which of the following graphs are graphs of functions. y

y

4

c. Find the income tax owed by a taxpayer whose adjusted gross income was $78,900.

a.

58.

y

−2

y

−2

x

2

2

x

4

−2

−2

4 2 2 −2 −4

y

59. x

−2

2

x

60.

y

4

−2

−2

V

2

F

2

−4 −2

c.

d.

y

−2

y

4

4

2

2

2

x

2

4

x

−2

−4

61. −4

−2

2 −2

4 x

−4

−2

2

62.

y

4 x

y

p

2

−2

2 s

−4

−4

−2

2 −2

4

x −2

2 −2

x

1.3 63.

64.

y

y 2

4 2

−2

t

x

2

Introduction to Functions

49

a. Find the length l of the shadow as a function of the distance d of the child from the lamppost. (Suggestion: Use the fact that the ratios of corresponding sides of similar triangles are equal.) b. What is the domain of the function?

−4

−2

65.

2

x

−3

66.

y

m

y

4 2

–2

r 2

–2 –2

4 x

4 x

2 k

–4

–2

67. Use the horizontal line test to determine which of the following functions are one-to-one. f as shown in Exercise 57 g as shown in Exercise 58 F as shown in Exercise 59 V as shown in Exercise 60 p as shown in Exercise 61

c. What is the length of the shadow when the child is 8 feet from the base of the lamppost? 71. DEPRECIATION A bus was purchased for $80,000. Assuming the bus depreciates at a rate of $6500 per year (straightline depreciation) for the first 10 years, write the value v of the bus as a function of the time t (measured in years) for 0 t 10. 72. DEPRECIATION A boat was purchased for $44,000. Assuming the boat depreciates at a rate of $4200 per year (straight-line depreciation) for the first 8 years, write the value v of the boat as a function of the time t (measured in years) for 0 t 8. 73. COST, REVENUE, AND PROFIT A manufacturer produces a product at a cost of $22.80 per unit. The manufacturer has a fixed cost of $400.00 per day. Each unit retails for $37.00. Let x represent the number of units produced in a 5-day period. a. Write the total cost C as a function of x.

68. Use the horizontal line test to determine which of the following functions are one-to-one. s as shown in Exercise 62 t as shown in Exercise 63 m as shown in Exercise 64 r as shown in Exercise 65 k as shown in Exercise 66

b. Write the revenue R as a function of x. c. Write the profit P as a function of x. (Hint: The profit function is given by P共x兲苷 R共x兲 C共x兲.)

69. GEOMETRY A rectangle has a length of l feet and a perimeter of 50 feet.

74. VOLUME OF A BOX An open box is to be made from a square piece of cardboard having dimensions 30 inches by 30 inches by cutting out squares of area x 2 from each corner, as shown in the figure.

a. Write the width w of the rectangle as a function of its length.

x

b. Write the area A of the rectangle as a function of its length.

30 – 2x

70. LENGTH OF A SHADOW A child 4 feet tall is standing near a street lamp that is 12 feet high. The light from the lamp casts a shadow as shown in the following diagram.

x x

30 – 2x

x

30 in.

a. Express the volume V of the box as a function of x. 12 ft 4 ft l

d

b. State the domain of V.

50

Chapter 1

Functions and Graphs

75. HEIGHT OF AN INSCRIBED CYLINDER A cone has an altitude of 15 centimeters and a radius of 3 centimeters. A right circular cylinder of radius r and height h is inscribed in the cone as shown in the figure. Use similar triangles to write h as a function of r.

78. TIME FOR A SWIMMER An athlete swims from point A to point B at a rate of 2 mph and runs from point B to point C at a rate of 8 mph. Use the dimensions in the figure to write the time t required to reach point C as a function of x. 3 mi x

?

B

15 cm

C

1 mi h A r 3 cm

76. VOLUME OF WATER Water is flowing into a conical drinking cup that has an altitude of 4 inches and a radius of 2 inches, as shown in 2 in. the figure. r a. Write the radius r of the surface of the water as a function of its depth h.

79. DISTANCE BETWEEN SHIPS At 12:00 noon Ship A is 45 miles due south of Ship B and is sailing north at a rate of 8 mph. Ship B is sailing east at a rate of 6 mph. Write the distance d between the ships as a function of the time t, where t 苷 0 represents 12:00 noon. Position of Ship B at noon

4 in.

b. Write the volume V of the water as a function of its depth h.

h

Ship A

d

77. DISTANCE FROM A BALLOON For the first minute of flight, a hot air balloon rises vertically at a rate of 3 meters per second. If t is the time in seconds that the balloon has been airborne, write the distance d between the balloon and a point on the ground 50 meters from the point of lift-off as a function of t.

Position of Ship B at time t

80. AREA A rectangle is bounded by the x- and y-axes and the 1 graph of y 苷 x 4. 2 y

4

(x, y)

2

d 2

4

6

8

x

50 m

a. Find the area of the rectangle as a function of x. b. Complete the following table.

1.3 x

51

Introduction to Functions

a. Find the area of the triangle as a function of x. (Suggestion: Let C be the point (0, 2) and D be the point (2, 0). Use the fact that ACP and PDB are similar triangles.)

Area

1 2

b. What is the domain of the function you found in a.?

4 6

83. LENGTH Two guy wires are attached to utility poles that are 40 feet apart, as shown in the following diagram.

7 c. What is the domain of this function? 81. AREA A piece of wire 20 centimeters long is cut at a point x centimeters from the left end. The left-hand piece is formed into the shape of a circle and the right-hand piece is formed into a square.

30 ft 20 ft

20 cm

x 40 ft

20 − x

x

a. Find the total length of the two guy wires as a function of x. a. Find the area enclosed by the two figures as a function of x.

b. Complete the following table. Round the length to the nearest hundredth of a foot. x

b. Complete the following table. Round the area to the nearest hundredth of a square centimeter. x

Total Length of Wires

0 10

Total Area Enclosed

20

0

30

4

40

8 12

c. What is the domain of this function?

16 84. SALES VS. PRICE A business finds that the number of feet f of pipe it can sell per week is a function of the price p in cents per foot as given by

20 c. What is the domain of this function? 82. AREA A triangle is bounded by the x- and y-axes and must pass through P共2, 2兲, as shown below. y 5 A(0, y) 4

f 共 p兲苷

320,000 , p 25

40 p 90

Complete the following table by evaluating f (to the nearest hundred feet) for the indicated values of p. p

40

50

60

75

90

f (p)

3 P(2, 2)

2 1

B(x, 0) 1

2

3

4

5

x

85. MODEL YIELD The yield Y of apples per tree is related to the amount x of a particular type of fertilizer applied (in pounds per year) by the function Y共x兲苷 400关1 5共x 1兲2 兴,

5 x 20

52

Chapter 1

Functions and Graphs

Complete the following table by evaluating Y (to the nearest apple) for the indicated amounts of fertilizer. 5

x

10

12.5

15

20

89. Determine whether 1 is in the range of f 共x兲苷

x1 . x1

90. Determine whether 0 is in the range of g共x兲苷

1 . x3

Y (x ) In Exercises 91 to 96, use a graphing utility.

86. MODEL COST A manufacturer finds that the cost C in dollars of producing x items of a product is given by C共x兲苷共 225 1.4兹x 兲 , 2

100 x 1000

Complete the following table by evaluating C (to the nearest dollar) for the indicated numbers of items. 100

x

200

500

750

1000

C (x)

91. Graph f 共x兲苷

冀x冁 for 4.7 x 4.7 and x 苷 0. 兩x兩

92. Graph f 共x兲苷

冀2x冁 for 4 x 4 and x 苷 0. 兩x兩

93. Graph: f 共x兲苷 x 2 2兩x兩 3 94. Graph: f 共x兲苷 x 2 兩2x 3兩

87. If f 共x兲苷 x 2 x 5 and f 共c兲苷 1, find c.

95. Graph: f 共x兲苷兩x 2 1兩兩x 2兩

88. If g共x兲苷 2x 2 4x 1 and g共c兲苷 4, find c.

96. Graph: f 共x兲苷兩x 2 2x兩 3

Connecting Concepts The notation f (x)|ba is used to denote the difference f (b) f (a). That is,

102. Given g共x, y兲苷 2x 2 兩y兩 3, find a. g共3, 4兲

b. g共1, 2兲

c. g共0, 5兲

d. g

e. g共c, 3c兲, c 0

f. g共c 5, c 2兲, c 0

f (x)| f (b) f (a) b a

In Exercises 97 to 100, evaluate f (x)|ba for the given function f and the indicated values of a and b.

97. f 共x兲苷 x 2 x; f 共x兲兩32

冉

A共a, b, c兲苷兹s共s a兲共s b兲共s c兲

99. f 共x兲苷 2x 3 3x 2 x; f 共x兲兩20

where s is the semiperimeter

100. f 共x兲苷兹8 x; f 共x兲兩80

s苷 In Exercises 101 to 104, each function has two or more independent variables.

101. Given f 共x, y兲苷 3x 5y 2, find

d. f 共4, 4兲

b. f 共0, 3兲 e. f 共k, 2k兲

冊

103. AREA OF A TRIANGLE The area of a triangle with sides a, b, and c is given by the function

98. f 共x兲苷 3x 2; f 共x兲兩74

a. f 共1, 7兲

1 1 , 2 4

c. f 共2, 4兲 f. f 共k 2, k 3兲

abc 2

Find A共5, 8, 11兲. 104. COST OF A PAINTER The cost in dollars to hire a house painter is given by the function C共h, g兲苷 15h 14g where h is the number of hours it takes to paint the house and g is the number of gallons of paint required to paint the house. Find C共18, 11兲.

1.4

Properties of Graphs

53

A fixed point of a function is a number a such that f (a) a. In Exercises 105 and 106, find all fixed points for the given function.

In Exercises 107 and 108, sketch the graph of the piecewisedefined function.

105. f 共x兲苷 x 2 3x 3

107. s共x兲苷

106. g共x兲苷

x x5

108. v共x兲苷

再再

1, 2,

if x is an integer if x is not an integer

2x 2, 1,

if x 苷 3 if x 苷 3

Projects 1.

DAY OF THE WEEK A formula known as Zeller’s Congruence makes use of the greatest integer function 冀x冁 to determine the day of the week on which a given day fell or will fall. To use Zeller’s Congruence, we first compute the integer z given by z苷

决13m5 1冴决 y4 冴决 4c 冴 d y 2c

a. Verify that December 7, 1941 was a Sunday.

The variables c, y, d, and m are defined as follows: c 苷 the century y 苷 the year of the century d 苷 the day of the month m 苷 the month, using 1 for March, 2 for April, . . . , 10 for December. January and February are assigned the values 11 and 12 of the previous year.

Section 1.4 ■ ■ ■ ■ ■

Symmetry Even and Odd Functions Translations of Graphs Reflections of Graphs Compressing and Stretching of Graphs

For example, for the date September 12, 2001, we use c 苷 20, y 苷 1, d 苷 12, and m 苷 7. The remainder of z divided by 7 gives the day of the week. A remainder of 0 represents a Sunday, a remainder of 1 a Monday, . . . , and a remainder of 6 a Saturday.

b. Verify that January 1, 2010 will fall on a Friday. c. Determine on what day of the week Independence Day (July 4, 1776) fell. d. Determine on what day of the week you were born.

Properties of Graphs P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A3.

PS1. Graph f 共x兲苷 x 2 and g共x兲苷 x 2 1 on the same coordinate grid. How is the graph of g related to the graph of f ? [1.2] PS2. For f 共x兲苷

3x 4 , show that f 共3兲苷 f 共3兲. [1.3] x 1 2

PS3. For f 共x兲苷 2x 3 5x, show that f 共2兲苷 f 共2兲. [1.3] PS4. Let f 共x兲苷 x 2 and g共x兲苷 x 3. Find f 共a兲 g共a兲 for a 苷 2, 1, 0, 1, 2. [1.3] PS5. What is the midpoint of the line segment between P共a, b兲 and Q共a, b兲? [1.2] PS6. What is the midpoint of the line segment between P共a, b兲 and Q共a, b兲? [1.2]

54

Chapter 1

Functions and Graphs

Symmetry C

B A l

B′ A′

Figure 1.49

C′

The graph in Figure 1.49 is symmetric with respect to the line l. Note that the graph has the property that if the paper is folded along the dotted line l, the point A will coincide with the point A, the point B will coincide with the point B, and the point C will coincide with the point C. One part of the graph is a mirror image of the rest of the graph across the line l. A graph is symmetric with respect to the y-axis if, whenever the point given by 共x, y兲 is on the graph, then 共x, y兲 is also on the graph. The graph in Figure 1.50 is symmetric with respect to the y-axis. A graph is symmetric with respect to the x-axis if, whenever the point given by 共x, y兲 is on the graph, then 共x, y兲 is also on the graph. The graph in Figure 1.51 is symmetric with respect to the x-axis. y

y (x, y) x

−x

(− x, y) y

x

(x, y) y

y

x

x −y x

(x, −y)

Figure 1.50

Figure 1.51

Symmetry with respect to the y-axis

Symmetry with respect to the x-axis

Tests for Symmetry with Respect to a Coordinate Axis The graph of an equation is symmetric with respect to ■

the y-axis if the replacement of x with x leaves the equation unaltered.

■

the x-axis if the replacement of y with y leaves the equation unaltered.

QUESTION

Which of the graphs below, I, II, or III, is a. symmetric with respect to the x-axis? b. symmetric with respect to the y-axis?

y

I

II

x

ANSWER

y

III

x

a. III is symmetric with respect to the x-axis. b. I is symmetric with respect to the y-axis.

y

x

1.4 EXAMPLE 1

»

55

Properties of Graphs

Determine Symmetries of a Graph

y

Determine whether the graph of the given equation has symmetry with respect to either the x- or the y-axis.

4

a.

y 苷 x2 2

b.

x 苷兩y兩 2

Solution 2

−2

a.

The equation y 苷 x 2 2 is unaltered by the replacement of x with x. That is, the simplification of y 苷共x兲2 2 yields the original equation y 苷 x 2 2. Thus the graph of y 苷 x 2 2 is symmetric with respect to the y-axis. However, the equation y 苷 x 2 2 is altered by the replacement of y with y. That is, the simplification of y 苷 x 2 2, which is y 苷 x 2 2, does not yield the original equation y 苷 x 2 2. The graph of y 苷 x 2 2 is not symmetric with respect to the x-axis. See 1.52.

b.

The equation x 苷兩y兩 2 is altered by the replacement of x with x. That is, the simplification of x 苷兩y兩 2, which is x 苷兩y兩 2, does not yield the original equation x 苷兩y兩 2. This implies that the graph of x 苷兩y兩 2 is not symmetric with respect to the y-axis. However, the equation x 苷兩y兩 2 is unaltered by the replacement of y with y. That is, the simplification of x 苷兩y兩 2 yields the original equation x 苷兩y兩 2. The graph of x 苷兩y兩 2 is symmetric with respect to the x-axis. See Figure 1.53.

x

2

y 苷 x2 2

Figure 1.52

N M

y 4 2

−2

2

4

x

−2 −4

x 苷兩y兩 2

Figure 1.53

»

Try Exercise 14, page 64

P Q

Definition of Symmetry with Respect to a Point P′

A graph is symmetric with respect to a point Q if for each point P on the graph there is a point P on the graph such that Q is the midpoint of the line segment PP .

M′ N′

Figure 1.54 y (x, y)

x y

x

The graph in Figure 1.54 is symmetric with respect to the point Q. For any point P on the graph, there exists a point P on the graph such that Q is the midpoint of P P. When we discuss symmetry with respect to a point, we frequently use the origin. A graph is symmetric with respect to the origin if, whenever the point given by 共x, y兲 is on the graph, then 共x, y兲 is also on the graph. The graph in Figure 1.55 is symmetric with respect to the origin.

−y

Test for Symmetry with Respect to the Origin (− x, − y)

−x

Figure 1.55 Symmetry with respect to the origin

The graph of an equation is symmetric with respect to the origin if the replacement of x with x and of y with y leaves the equation unaltered.

56

Chapter 1

Functions and Graphs EXAMPLE 2

»

Determine Symmetry with Respect to the Origin

Determine whether the graph of each equation has symmetry with respect to the origin. xy 苷 4

a.

b.

y 苷 x3 1

Solution a.

The equation xy 苷 4 is unaltered by the replacement of x with x and y with y. That is, the simplification of 共x兲共y兲苷 4 yields the original equation xy 苷 4. Thus the graph of xy 苷 4 is symmetric with respect to the origin. See Figure 1.56.

b.

The equation y 苷 x 3 1 is altered by the replacement of x with x and y with y. That is, the simplification of y 苷共x兲3 1, which is y 苷 x 3 1, does not yield the original equation y 苷 x 3 1. Thus the graph of y 苷 x 3 1 is not symmetric with respect to the origin. See Figure 1.57. y

y 4

4 2 2 −4

−2

2

4

x

−2

−2

2

x

−2 y

−4

2

−2

2

x

xy 苷 4

y 苷 x3 1

Figure 1.56

Figure 1.57

»

Try Exercise 24, page 64

−2

兩x兩兩y兩苷 2

Figure 1.58

Some graphs have more than one symmetry. For example, the graph of 兩x兩兩y兩苷 2 has symmetry with respect to the x-axis, the y-axis, and the origin. Figure 1.58 is the graph of 兩x兩兩y兩苷 2.

Even and Odd Functions Some functions are classified as either even or odd. Definition of Even and Odd Functions The function f is an even function if f共x兲苷 f共x兲 for all x in the domain of f The function f is an odd function if f共x兲苷 f共x兲 for all x in the domain of f

1.4 EXAMPLE 3

»

57

Properties of Graphs

Identify Even or Odd Functions

Determine whether each function is even, odd, or neither. f共x兲苷 x 3

a.

b.

F共x兲苷兩x兩

c.

h共x兲苷 x 4 2x

Solution Replace x with x and simplify. a.

f共x兲苷共x兲 3 苷 x 3 苷共x 3 兲苷 f共x兲 Because f共x兲苷 f共x兲, this function is an odd function.

b.

F共x兲苷兩x兩苷兩x兩苷 F共x兲 Because F共x兲苷 F共x兲, this function is an even function.

c.

h共x兲苷共x兲4 2共x兲苷 x 4 2x This function is neither an even nor an odd function because h共x兲苷 x 4 2x, which is not equal to either h共x兲 or h共x兲.

»

Try Exercise 44, page 64

The following properties are results of the tests for symmetry: ■

The graph of an even function is symmetric with respect to the y-axis.

■

The graph of an odd function is symmetric with respect to the origin.

The graph of f in Figure 1.59 is symmetric with respect to the y-axis. It is the graph of an even function. The graph of g in Figure 1.60 is symmetric with respect to the origin. It is the graph of an odd function. The graph of h in Figure 1.61 is not symmetric with respect to the y-axis and is not symmetric with respect to the origin. It is neither an even nor an odd function.

y

y y

f

h

g

x

x x

Figure 1.59

Figure 1.60

Figure 1.61

The graph of an even function is symmetric with respect to the y-axis.

The graph of an odd function is symmetric with respect to the origin.

The graph of a function that is neither even nor odd is not symmetric with respect to the y-axis or the origin.

58

Chapter 1

Functions and Graphs

Translations of Graphs y

The shape of a graph may be exactly the same as the shape of another graph; only their positions in the xy-plane may differ. For example, the graph of y 苷 f共x兲 2 is the graph of y 苷 f共x兲 with each point moved up vertically 2 units. The graph of y 苷 f共x兲 3 is the graph of y 苷 f共x兲 with each point moved down vertically 3 units. See Figure 1.62. The graphs of y 苷 f共x兲 2 and y 苷 f共x兲 3 in Figure 1.62 are called vertical translations of the graph of y 苷 f共x兲.

5 y = f (x) + 2 −1

1

x

2

−5

Vertical Translations

y = f (x)

If f is a function and c is a positive constant, then the graph of

−10 y = f(x) − 3

Figure 1.62

y

y = h(x + 3)

2

y = h(x)

Figure 1.63

4

y 苷 f共x兲 c is the graph of y 苷 f共x兲 shifted up vertically c units.

■

y 苷 f共x兲 c is the graph of y 苷 f共x兲 shifted down vertically c units.

In Figure 1.63, the graph of y 苷 h共x 3兲 is the graph of y 苷 h共x兲 with each point shifted to the left horizontally 3 units. Similarly, the graph of y 苷 h共x 3兲 is the graph of y 苷 h共x兲 with each point shifted to the right horizontally 3 units. The graphs of y 苷 h共x 3兲 and y 苷 h共x 3兲 in Figure 1.63 are called horizontal translations of the graph of y 苷 h共x兲.

y = h(x − 3)

4

−4

■

x

Horizontal Translations If f is a function and c is a positive constant, then the graph of ■

y 苷 f共x c兲 is the graph of y 苷 f共x兲 shifted left horizontally c units.

■

y 苷 f共x c兲 is the graph of y 苷 f共x兲 shifted right horizontally c units.

Integrating Technology A graphing calculator can be used to draw the graphs of a family of functions. For instance, f共x兲苷 x 2 c constitutes a family of functions with parameter c. The only feature of the graph that changes is the value of c. A graphing calculator can be used to produce the graphs of a family of curves for specific values of the parameter. The LIST feature of the calculator can be used. For instance, to graph f共x兲苷 x 2 c for c 苷 2, 0, and 1, we will create a list and use that list to produce the family of curves. The keystrokes for a TI-83/TI-83 Plus/TI-84 Plus calculator are given below. 2nd { -2 , 0 , 1 2nd } STO 2nd L1 Now use the Y= key to enter Y= X x2 +

2nd

L1

ZOOM

6

1.4

59

Properties of Graphs

Sample screens for the keystrokes and graphs are shown here. You can use similar keystrokes for Exercises 75–82 of this section. {-2, 0, 1} –– > L1

Plot1

Plot2

Plot3

\Y1 = X2+L1 \Y2 = \Y3 = \Y4 = \Y5 = \Y6 = \Y7

{-2 0 1}

10

−10

10

−10

EXAMPLE 4

3

g共x兲苷 x 3 2

a.

f (x) = x 3 2

h共x兲苷共x 1兲3

b.

Solution

1

−1

Graph by Using Translations

Use vertical and horizontal translations of the graph of f共x兲苷 x 3, shown in Figure 1.64, to graph

y

−2

»

1 −1

2

a.

The graph of g共x兲苷 x 3 2 is the graph of f共x兲苷 x 3 shifted down vertically 2 units. See Figure 1.65.

b.

The graph of h共x兲苷共x 1兲3 is the graph of f共x兲苷 x 3 shifted to the left horizontally 1 unit. See Figure 1.66.

x

−2

y

y

−3

3

3

2

2

1

h(x) = (x + 1)3 1

Figure 1.64

−1

−2 f(x) = x

3

1

x

2

−1

1 −1

−1 3

g(x) = x − 2 −3

Figure 1.65

»

−2

f(x) = x 3

Try Exercise 58, page 65

−2 −3

Figure 1.66

2

x

60

Chapter 1

Functions and Graphs

Reflections of Graphs

y

The graph of y 苷 f共x兲 cannot be obtained from the graph of y 苷 f共x兲 by a combination of vertical and/or horizontal shifts. Figure 1.67 illustrates that the graph of y 苷 f共x兲 is the reflection of the graph of y 苷 f共x兲 across the x-axis. The graph of y 苷 f共x兲 is the reflection of the graph of y 苷 f共x兲 across the y-axis, as shown in Figure 1.68.

y = f(x) x y = − f (x)

Figure 1.67

Reflections The graph of

y y = f (−x)

y = f (x)

■

y 苷 f共x兲 is the graph of y 苷 f共x兲 reflected across the x-axis.

■

y 苷 f共x兲 is the graph of y 苷 f共x兲 reflected across the y-axis.

x

Figure 1.68

EXAMPLE 5

y

2

−1

g共x兲苷共兹x 1 1 兲

a.

1 −3 − 2 − 1

1

2

Graph by Using Reflections

Use reflections of the graph of f共x兲苷兹x 1 1, shown in Figure 1.69, to graph

f

3

»

3

h共x兲苷兹x 1 1

b.

Solution

x

−2

a.

Because g共x兲苷 f共x兲, the graph of g is the graph of f reflected across the x-axis. See Figure 1.70.

b.

Because h共x兲苷 f共x兲, the graph of h is the graph of f reflected across the y-axis. See Figure 1.71.

−3

Figure 1.69

y

y f

3

− 3 − 2 −1

2

1

1

−1

1

2

4

−3 y = −f (x) + 3

Figure 1.70

−2

−4

Figure 1.72

−3 −2 −1

−1

1

2

3

x

−2 g

−3

Figure 1.71

Try Exercise 68, page 66

4

y = f (x)

x

»

2

−4

3

f

3

2

−2 y

h

x

Some graphs of functions can be constructed by using a combination of translations and reflections. For instance, the graph of y 苷 f共x兲 3 in Figure 1.72 was obtained by reflecting the graph of y 苷 f共x兲 in Figure 1.72 across the x-axis and then shifting that graph up vertically 3 units.

1.4

61

Properties of Graphs

Compressing and Stretching of Graphs y 4

y = | x|

2 y=

1 |x| 2 −2

x

2

Figure 1.73

The graph of the equation y 苷 c f共x兲 for c 苷 1 vertically compresses or stretches the graph of y 苷 f共x兲. To determine the points on the graph of y 苷 c f共x兲, multiply the y-coordinate of each point on the graph of y 苷 f共x兲 by c. For example, 1 Figure 1.73 shows that the graph of y 苷兩x兩 can be obtained by plotting points 2 that have a y-coordinate that is one-half of the y-coordinate of those points that make up the graph of y 苷兩x兩. If 0 c 1, then the graph of y 苷 c f共x兲 is obtained by compressing the graph of y 苷 f共x兲. Figure 1.73 illustrates the vertical compressing of the graph of y 苷兩x兩 1 y y = 2| x| toward the x-axis to form the graph of y 苷兩x兩. 2 4 If c 1, then the graph of y 苷 c f共x兲 is obtained by stretching the graph of y 苷 f共x兲. For example, if f共x兲苷兩x兩, then we obtain the graph of 2 y = | x|

y 苷 2f共x兲苷 2兩x兩 by stretching the graph of f away from the x-axis. See Figure 1.74.

−2

2

Figure 1.74

Vertical Stretching and Compressing of Graphs If f is a function and c is a positive constant, then ■

if c 1, the graph of y 苷 c f共x兲 is the graph of y 苷 f共x兲 stretched vertically away from the x-axis by a factor of c.

■

if 0 c 1, the graph of y 苷 c f共x兲 is the graph of y 苷 f共x兲 compressed vertically toward the x-axis by a factor of c.

EXAMPLE 6

Graph: H共x兲苷

(− 4, 4) y=

1 |x| 4 (− 4, 1)

(4, 4)

4 2

(4, 1)

−4

4 (− 4, − 2)

y=

1 |x| − 3 4

(4, − 2)

−2

(0, − 3)

Figure 1.75

Graph by Using Vertical Compressing and Shifting

1 兩x兩 3 4

Solution

y = | x|

y

»

x

The graph of y 苷兩x兩 has a V shape that has its lowest point at 共0, 0兲 and 1 passes through 共4, 4兲 and 共4, 4兲. The graph of y 苷兩x兩 is a compression of 4 the graph of y 苷兩x兩. The y-coordinates of the ordered pairs 共0, 0兲, 共4, 1兲, and 共4, 1兲 are obtained by multiplying the y-coordinates of the ordered pairs 1 共0, 0兲, 共4, 4兲, and 共4, 4兲 by . To find the points on the graph of H, we still 4 need to subtract 3 from each y-coordinate. Thus the graph of H is a V shape that has its lowest point at 共0, 3兲 and passes through 共4, 2兲 and 共4, 2兲. See Figure 1.75.

»

Try Exercise 70, page 66

x

62

Chapter 1

Functions and Graphs Some functions can be graphed by using a horizontal compressing or stretching of a given graph. The procedure makes use of the following concept.

Horizontal Compressing and Stretching of Graphs If f is a function and c is a positive constant, then ■

if c 1, the graph of y 苷 f共c x兲 is the graph of y 苷 f共x兲 compressed 1 horizontally toward the y-axis by a factor of . c

■

if 0 c 1, the graph of y 苷 f共c x兲 is the graph of y 苷 f共x兲 stretched 1 horizontally away from the y-axis by a factor of . c

If the point 共x, y兲 is on the graph of y 苷 f共x兲, then the graph of y 苷 f共cx兲 will 1 contain the point x, y . c

冉冊

EXAMPLE 7

»

Graph by Using Horizontal Compressing and Stretching

Use the graph of y 苷 f共x兲, shown in Figure 1.76, to graph a.

y 苷 f共2x兲

b.

y苷f

冉冊 1 x 3

y 2

−6

−2

y = f(x)

2

4

6

x

−2

Figure 1.76

Solution a.

Because 2 1, the graph of y 苷 f共2x兲 is a horizontal compression of the 1 graph of y 苷 f共x兲 by a factor of . For example, the point 共2, 0兲 on the 2 graph of y 苷 f共x兲 becomes the point 共1, 0兲 on the graph of y 苷 f共2x兲. See Figure 1.77.

b.

冉冊

1 1 1, the graph of y 苷 f x is a horizontal stretching of 3 3 the graph of y 苷 f共x兲 by a factor of 3. For example, the point 共1, 1兲 on the Since 0

1.4

63

Properties of Graphs

graph of y 苷 f共x兲 becomes the point 共3, 1兲 on the graph of y 苷 f See Figure 1.78. y

冉冊

1 x . 3

y

2

y = f (2x)

y=f

2

1

( 3 x)

−6 2

4

6

x

−6

−2

Figure 1.77

−2

2

4

6

x

−2

Figure 1.78

»

Try Exercise 72, page 66

Topics for Discussion 1. Discuss the meaning of symmetry of a graph with respect to a line. How do you determine whether a graph has symmetry with respect to the x-axis? with respect to the y-axis? 2. Discuss the meaning of symmetry of a graph with respect to a point. How do you determine whether a graph has symmetry with respect to the origin? 3. What does it mean to reflect a graph across the x-axis or across the y-axis? 4. Explain how the graphs of y 苷 2x 3 x 2 and y 苷 2共x兲3 共x兲2 are related. 5. Given the graph of the function y 苷 f共x兲, explain how to obtain the graph of the function y 苷 f共x 3兲 1. 6. The graph of the step function y 苷冀x冁 has steps that are 1 unit wide. Determine how wide the steps are for the graph of y 苷

决 13 x冴.

64

Chapter 1

Functions and Graphs

Exercise Set 1.4 In Exercises 1 to 6, plot the image of the given point with respect to a. the y-axis. Label this point A. b. the x-axis. Label this point B. c. the origin. Label this point C.

In Exercises 13 to 21, determine whether the graph of each equation is symmetric with respect to the a. x-axis, b. y-axis.

13. y 苷 2x 2 5

14. x 苷 3y 2 7

15. y 苷 x 3 2

1. P共5, 3兲

2. Q共4, 1兲

3. R共2, 3兲

16. y 苷 x 5 3x

17. x 2 y 2 苷 9

18. x 2 y 2 苷 10

4. S共5, 3兲

5. T共4, 5兲

6. U共5, 1兲

19. x 2 苷 y 4

20. xy 苷 8

21. 兩x兩兩y兩苷 6

In Exercises 7 and 8, sketch a graph that is symmetric to the given graph with respect to the x-axis.

7.

8.

y

−4

y

4

4

2

2

−2

2

4 x

−4

−2

1

x

In Exercises 22 to 30, determine whether the graph of each equation is symmetric with respect to the origin.

22. y 苷 x 1

23. y 苷 3x 2

24. y 苷 x 3 x

25. y 苷 x 3

26. y 苷

9 x

27. x 2 y 2 苷 10

28. x 2 y 2 苷 4

29. y 苷

x 兩x兩

30. 兩y兩苷兩x兩

−2 −3

In Exercises 31 to 42, graph the given equation. Label each intercept. Use the concept of symmetry to confirm that the graph is correct.

In Exercises 9 and 10, sketch a graph that is symmetric to the given graph with respect to the y-axis.

9.

10.

y

31. y 苷 x 2 1

32. x 苷 y 2 1

y

33. y 苷 x 3 x

34. y 苷 x 3

4

35. xy 苷 4

36. xy 苷 8

37. y 苷 2兩x 4兩

38. y 苷兩x 2兩 1

39. y 苷共x 2兲2 4

40. y 苷共x 1兲2 4

41. y 苷 x 兩x兩

42. 兩y兩苷兩x兩

6 4 2 −2

2

4

−2

6 x

2

x

−2

−2

−4

In Exercises 43 to 56, determine whether the given function is an even function, an odd function, or neither. In Exercises 11 and 12, sketch a graph that is symmetric to the given graph with respect to the origin.

43. g共x兲苷 x 2 7

44. h共x兲苷 x 2 1

11.

45. F共x兲苷 x 5 x 3

46. G共x兲苷 2x 5 10

47. H共x兲苷 3兩x兩

48. T共x兲苷兩x兩 2

49. f 共x兲苷 1

50. k共x兲苷 2 x x 2

51. r共x兲苷兹x 2 4

52. u共x兲苷兹3 x 2

12.

y

y

4

2

2

2

4

x

−4

−2

x

1.4 53. s共x兲苷 16x 2

54. v共x兲苷 16x 2 x 3

55. w共x兲苷 4 兹 x

56. z共x兲苷

x3 x 1 2

Properties of Graphs

65

61. Let f be a function such that f 共2兲苷 5, f 共0兲苷 2, and f 共1兲苷 0. Give the coordinates of three points on the graph of a. y 苷 f 共x 3兲

b. y 苷 f 共x兲 1

57. Use the graph of f to sketch the graph of a. y 苷 f 共x兲 3

b. y 苷 f 共x 3兲 y f

2

−2

62. Let g be a function such that g共3兲苷 1, g共1兲苷 3, and g共4兲苷 2. Give the coordinates of three points on the graph of a. y 苷 g共x 2兲

63. Use the graph of f to sketch the graph of

x

2

b. y 苷 g共x兲 2

a. y 苷 f 共x兲

b. y 苷 f 共x兲

58. Use the graph of g to sketch the graph of a. y 苷 g共x兲 2

y

b. y 苷 g共x 3兲

3

y 2

4

1

2

−4

−2

f

g −4

4 x

2

−2

2

4

x

−1

−2 −4

64. Use the graph of g to sketch the graph of

59. Use the graph of f to sketch the graph of a. y 苷 f 共x 2兲

b. y 苷 f 共x兲 2

a. y 苷 g共x兲

b. y 苷 g共x兲

y

y 4

3 f

2

g

2

1 −4

−2

2

4

−4

x

2

4

x

−2

−1

−4

60. Use the graph of g to sketch the graph of a. y 苷 g共x 1兲

−2

b. y 苷 g共x兲 1 y 4

65. Let f be a function such that f 共1兲苷 3 and f 共2兲苷 4. Give the coordinates of two points on the graph of

g

a. y 苷 f 共x兲

3

b. y 苷 f 共x兲

2

66. Let g be a function such that g共4兲苷 5 and g共3兲苷 2. Give the coordinates of two points on the graph of

1 −4

−2

2

4

x

a. y 苷 g共x兲

b. y 苷 g共x兲

66

Chapter 1

Functions and Graphs

67. Use the graph of F to sketch the graph of a. y 苷 F共x兲

b. y 苷 F共x兲 y

71. Use the graph of y 苷 f 共x兲 to sketch the graph of a. y 苷 f 共2x兲

b. y 苷 f

F

冉冊 1 x 3

y

2

2 −2

1

4 x

2

−8 −6 −4 −2 −1

f 2

4

6

8

x

68. Use the graph of E to sketch the graph of a. y 苷 E共x兲

b. y 苷 E共x兲 y

72. Use the graph of y 苷 g共x兲 to sketch the graph of

冉冊

4

a. y 苷 g共2x兲

2

−4

b. y 苷 g

E

−2

y

4 x

2

1 x 2

2

g

−2 −4

−8 −6 −4

69. Use the graph of m共x兲苷 x 2 2x 3 to sketch the graph 1 of y 苷 m共x兲 3. 2 y

−2

2

6

8

x

−2

73. Use the graph of y 苷 h共x兲 to sketch the graph of

冉冊

a. y 苷 h共2x兲

m

4

b. y 苷 h

2

1 x 2

y 2

x

2

h

1 −2 π

−4

−π

2π x

π

−1 −2

70. Use the graph of n共x兲苷 x 2 2x 8 to sketch the graph 1 of y 苷 n共x兲 1. 2

74. Use the graph of y 苷 j共x兲 to sketch the graph of

冉冊

y

a. y 苷 j共2x兲

10

b. y 苷 j

8 n

y

6

2

4 −2π 2 −2

j

1

2 −2

1 x 3

−π

π

x −2

2π x

1.4

67

Properties of Graphs

80. On the same coordinate axes, graph

In Exercises 75 to 82, use a graphing utility.

M共x兲苷 c兹x 2 4 75. On the same coordinate axes, graph

for c 苷 1,

3

G共x兲苷兹 x c for c 苷 0, 1, and 3.

1 , and 3. 3

81. On the same coordinate axes, graph S共x兲苷 c共兩x 1兩兩x兩兲

76. On the same coordinate axes, graph 3

H共x兲苷兹 x c

for c 苷 1,

for c 苷 0, 1, and 3.

1 , and 4. 4

82. On the same coordinate axes, graph

冉冊

77. On the same coordinate axes, graph

T共x兲苷 c

J共x兲苷兩2共x c兲 3兩兩x c兩 for c 苷 0, 1, and 2.

for c 苷 1,

78. On the same coordinate axes, graph K共x兲苷兩x 1兩兩x兩 c

2 3 , and . 3 2

83. Graph V共x兲苷冀cx冁, 0 x 6, for each value of c.

for c 苷 0, 1, and 2.

a. c 苷 1

b. c 苷

79. On the same coordinate axes, graph

1 2

c. c 苷 2

84. Graph W共x兲苷冀cx冁 cx, 0 x 6, for each value of c.

L共x兲苷 cx 2 for c 苷 1,

x 兩x兩

1 , and 2. 2

b. c 苷

a. c 苷 1

1 3

c. c 苷 3

Connecting Concepts 85. Use the graph of f 共x兲苷

2 to determine an equation x2 1

86. Use the graph of f 共x兲苷 x 兹2 x to determine an equation for each graph.

for each graph.

y y 2

4 f f

2

−2

y

a.

x

2

b.

−2

y

a.

2 2

b.

2 2

−2

y 4

2

x

y 4 2

x 2

x −2

2

4

x

2

−2

−2

−4

−4

4 x

68

Chapter 1

Functions and Graphs

Projects 1. DIRICHLET FUNCTION We owe our present-day definition of a function to the German mathematician Peter Gustav Dirichlet (1805 – 1859). He created the following unusual function, which is now known as the Dirichlet function. f 共x兲苷

再

0, 1,

3.

f 共x兲苷

a. What is its domain? b. What is its range?

共x 2兲共x 1兲共x 2兲

is a line with a y-intercept of 1, a slope of 1, and a hole at 共2, 3兲. Use a graphing utility to graph f. Explain why a graphing utility might not show the hole at 共2, 3兲.

if x is a rational number if x is an irrational number

Answer the following questions about the Dirichlet function.

A LINE WITH A HOLE The graph of the function

4.

FINDING A COMPLETE GRAPH Use a graphing utility to graph the function f 共x兲苷 3x 5/3 6x 4/3 2 for 2 x 10. Compare your graph with the graph below. Does your graph include the part to the left of the y-axis? If not, how might you enter the function so that the graphing utility would include this part? y

c. What are its x-intercepts? d. What is its y-intercept?

2

e. Is it an even or an odd function?

4

6

8

x

−4

f.

g.

2.

Explain why a graphing calculator cannot be used to produce an accurate graph of the Dirichlet function.

−8

Write a sentence or two that describes the graph of the Dirichlet function.

−16

ISOLATED POINT Consider the function given by the equation y 苷兹共x 1兲2共x 2兲 1 Verify that the point 共1, 1兲 is a solution of the equation. Now use a graphing utility to graph the function. Does your graph include the isolated point at 共1, 1兲, as shown below? If the graphing utility you used failed to include the point 共1, 1兲, explain at least one reason for the omission of this isolated point. y 5

y=

(x − 1)2(x − 2) + 1

4 3 2 1 1

2

3

x

−12

f(x) = 3x5/3 − 6x4/3 + 2

1.5

Section 1.5 ■ ■ ■

Operations on Functions The Difference Quotient Composition of Functions

The Algebra of Functions

69

The Algebra of Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A5.

PS1. If f 共x兲苷 x 2 3x 1 and g共x兲苷 4x 5, find f 共3兲 g共3兲. [1.3] PS2. If f 共x兲苷 3x2 x 4 and g共x兲苷 2x 5, find f 共2兲 g共2兲. [1.3] In Exercises PS3 and PS4, find each of the following for f (x) 2x 2 5x 2.

PS3. f 共3a兲 [1.3]

PS4. f 共2 h兲 [1.3]

In Exercises PS5 and PS6, find the domain of each function.

PS5. F共x兲苷

x [1.3] x1

PS6. r共x兲苷兹2x 8 [1.3]

Operations on Functions Functions can be defined in terms of other functions. For example, the function defined by h共x兲苷 x 2 8x is the sum of f共x兲苷 x 2

and

g共x兲苷 8x

Thus, if we are given any two functions f and g, we can define the four new funcf tions f g, f g, fg, and as follows. g Definitions of Operations on Functions If f and g are functions with domains Df and Dg, then we define the sum, difference, product, and quotient of f and g as Sum

共 f g兲共x兲苷 f共x兲 g共x兲

Domain: Df Dg

Difference

共 f g兲共x兲苷 f共x兲 g共x兲

Domain: Df Dg

Product

共 f g兲共x兲苷 f共x兲 g共x兲

Domain: Df Dg

Quotient

冉冊

f f共x兲共x兲苷 g g共x兲

Domain: Df Dg, g共x兲苷 0

Example

Let f共x兲苷 3x 2 and g共x兲苷 x 2 6. Then 共 f g兲共x兲苷 f共x兲 g共x兲苷共3x 2兲共x 2 6兲苷 x 2 3x 4 共 f g兲共x兲苷 f共x兲 g共x兲苷共3x 2兲共x 2 6兲苷 x 2 3x 8 共 f g兲共x兲苷 f共x兲 g共x兲苷共3x 2兲共x 2 6兲苷 3x 3 2x 2 18x 12

冉冊

f f (x) 3x 2 共x兲苷苷 2 g g(x) x 6

70

Chapter 1

Functions and Graphs EXAMPLE 1

»

Determine the Domain of a Function

If f共x兲苷兹x 1 and g共x兲苷 x 2 4, find the domains of f g, f g, fg, f and . g

Solution Note that f has the domain 兵x 兩 x 1其 and g has the domain of all real numbers. Therefore, the domain of f g, f g, and fg is 兵x 兩 x 1其. Because f g共x兲苷 0 when x 苷 2 or x 苷 2, neither 2 nor 2 is in the domain of . g f The domain of is 兵x 兩 x 1 and x 苷 2其. g

»

Try Exercise 10, page 77

EXAMPLE 2

»

Evaluate Functions

Let f共x兲苷 x 2 9 and g共x兲苷 2x 6. Find 共 f g兲共5兲

a.

b.

共 fg兲共1兲

c.

冉冊

f 共4兲 g

Solution a.

共 f g兲共x兲苷 f共x兲 g共x兲苷共x 2 9兲共2x 6兲苷 x 2 2x 3 Therefore, 共 f g兲共5兲苷共5兲2 2共5兲 3 苷 25 10 3 苷 32.

b.

共 fg兲共x兲苷 f共x兲 g共x兲苷共x 2 9兲共2x 6兲苷 2x 3 6x 2 18x 54 Therefore, 共 fg兲共1兲苷 2共1兲3 6共1兲2 18共1兲 54 苷 2 6 18 54 苷 32.

冉冊

f f共x兲 x2 9 共x 3兲共x 3兲 x3 共x兲苷苷苷苷 , g g共x兲 2x 6 2共x 3兲 2 f 43 1 Therefore, 共4兲苷苷 . g 2 2

c.

冉冊

x 苷 3

»

Try Exercise 14, page 77

兰

Calculus Connection

The Difference Quotient The expression f共x h兲 f共x兲 , h

h苷0

is called the difference quotient of f. It enables us to study the manner in which a function changes in value as the independent variable changes.

1.5 EXAMPLE 3

»

The Algebra of Functions

71

Determine a Difference Quotient

Determine the difference quotient of f共x兲苷 x 2 7.

Solution f共x h兲 f共x兲关共x h兲2 7兴关x 2 7兴苷 h h

• Apply the difference quotient.

关x 2 2xh h2 7兴关x 2 7兴 h x 2 2xh h2 7 x 2 7 苷 h 2 2xh h h共2x h兲苷苷苷 2x h h h 苷

»

Try Exercise 30, page 77

y

The difference quotient 2x h of f共x兲苷 x 2 7 from Example 3 is the slope of the secant line through the points

l2 m=3

14

共x, f共x兲兲

12

(2, 11)

10 f 8

(1, 8)

(1.1, 8.21)

l1 m = 2.1

and

共x h, f共x h兲兲

For instance, let x 苷 1 and h 苷 1. Then the difference quotient is 2x h 苷 2共1兲 1 苷 3

4

This is the slope of the secant line l2 through 共1, 8兲 and 共2, 11兲, as shown in Figure 1.79. If we let x 苷 1 and h 苷 0.1, then the difference quotient is

2

2x h 苷 2共1兲 0.1 苷 2.1

6

1

2

Figure 1.79

x

This is the slope of the secant line l1 through 共1, 8兲 and 共1.1, 8.21兲. The difference quotient f共x h兲 f共x兲 h

s(t)

Figure 1.80

can be used to compute average velocities. In such cases it is traditional to replace f with s (for distance), the variable x with the variable a (for the time at the start of an observed interval of time), and the variable h with t (read as “delta t”), where t is the difference between the time at the end of an interval and the time at the start of the interval. For example, if an experiment is observed over the time interval from t 苷 3 seconds to t 苷 5 seconds, then the time interval is denoted as 关3, 5兴 with a 苷 3 and t 苷 5 3 苷 2. Thus if the distance traveled by a ball that rolls down a ramp is given by s共t兲, where t is the time in seconds after the ball is released (see Figure 1.80), then the average velocity of the ball over the interval t 苷 a to t 苷 a t is the difference quotient s共a t兲 s共a兲 t

72

Chapter 1

Functions and Graphs EXAMPLE 4

»

Evaluate Average Velocities

The distance traveled by a ball rolling down a ramp is given by s共t兲苷 4t 2, where t is the time in seconds after the ball is released, and s共t兲 is measured in feet. Evaluate the average velocity of the ball for each time interval. 关3, 5兴

a.

b.

关3, 4兴

c.

关3, 3.5兴

d.

关3, 3.01兴

Solution In this case, a 苷 3 and t 苷 5 3 苷 2. Thus the average velocity over this interval is

a.

s共a t兲 s共a兲 s共3 2兲 s共3兲 s共5兲 s共3兲 100 36 苷苷苷 t 2 2 2 苷 32 feet per second Let a 苷 3 and t 苷 4 3 苷 1.

b.

s共a t兲 s共a兲 s共3 1兲 s共3兲 s共4兲 s共3兲 64 36 苷苷苷 t 1 1 1 苷 28 feet per second Let a 苷 3 and t 苷 3.5 3 苷 0.5.

c.

s共a t兲 s共a兲 s共3 0.5兲 s共3兲 49 36 苷苷苷 26 feet per second t 0.5 0.5 d. Let a 苷 3 and t 苷 3.01 3 苷 0.01. s共a t兲 s共a兲 s共3 0.01兲 s共3兲 36.2404 36 苷苷 t 0.01 0.01 苷 24.04 feet per second

»

Try Exercise 72, page 79

Composition of Functions Composition of functions is another way in which functions can be combined. This method of combining functions uses the output of one function as the input for a second function. Suppose that the spread of oil from a leak in a tanker can be approximated by a circle with the tanker at its center. The radius r (in feet) of the spill t hours after the leak begins is given by r共t兲苷 150兹t. The area of the spill is the area of a circle and is given by the formula A共r兲苷 pr 2. To find the area of the spill 4 hours after the leak begins, we first find the radius of the spill and then use that number to find the area of the spill. r共t兲苷 150兹t r共4兲苷 150兹4 苷 150共2兲苷 300

• t 4 hours

A共r兲苷 pr 2 A共300兲苷 p共300 2 兲

• r 300 feet

苷 90,000p ⬇ 283,000

The area of the spill after 4 hours is approximately 283,000 square feet.

1.5

The Algebra of Functions

73

There is an alternative way to solve this problem. Because the area of the spill depends on the radius and the radius depends on the time, there is a relationship between area and time. We can determine this relationship by evaluating the formula for the area of a circle using r共t兲苷 150兹t. This will give the area of the spill as a function of time. A共r兲苷 pr 2 A关r共t兲兴苷 p关r共t兲兴 2 2 苷 p 关 150兹t 兴 A共t兲苷 22,500pt

• Replace r by r(t). • r(t) 150兹t • Simplify.

The area of the spill as a function of time is A共t兲苷 22,500pt. To find the area of the oil spill after 4 hours, evaluate this function at t 苷 4. A共t兲苷 22,500pt A共4兲苷 22,500p共4兲

• t 4 hours

苷 90,000p ⬇ 283,000 This is the same result we calculated earlier. The function A共t兲苷 22,500pt is referred to as the composition of A with r. The notation A ⴰ r is used to denote this composition of functions. That is, 共A ⴰ r兲共t兲苷 22,500pt

take note The requirement in the definition of the composition of two functions that g共x兲 be in the domain of f for

Definition of the Composition of Two Functions Let f and g be two functions such that g共x兲 is in the domain of f for all x in the domain of g. Then the composition of the two functions, denoted by f ⴰ g, is the function whose value at x is given by 共 f ⴰ g兲共x兲苷 f 关 g共x兲兴.

all x in the domain of g is important. For instance, let f共x兲苷

1 x1

and

g共x兲苷 3x 5 When x 苷 2, g共2兲苷 3共2兲 5 苷 1 f 关g共2兲兴苷 f共1兲 1 苷 11 1 苷 0

• Undefined

In this case, g共2兲 is not in the domain of f. Thus the composition 共f ⴰ g兲共x兲 is not defined at 2.

The function defined by 共 f ⴰ g兲共x兲 is also called the composite of f and g. We read 共 f ⴰ g兲共x兲 as “f circle g of x” and f 关 g共x兲兴 as “f of g of x.” Consider the functions f共x兲苷 2x 1 and g共x兲苷 x 2 3. The expression 共 f ⴰ g兲共1兲 (or, equivalently, f 关 g共1兲兴) means to evaluate the function f at g共1兲. g共x兲苷 x 2 3 g共1兲苷共1兲2 3 g共1兲苷 2 f共x兲苷 2x 1 f [g共1兲] 苷 2[g共1兲] 1 f共2兲苷 2共2兲 1 苷 5

• Evaluate g at 1.

• Replace x by g(1). • g(1) 2

A graphical depiction of the composition 共 f ⴰ g兲共1兲 would look something like Figure 1.81 on page 74.

74

Chapter 1

Functions and Graphs −1

Square Subtract 3

−2

g(x) = x2 − 3 Multiply by 2 Subtract 1

−5

f(x) = 2x − 1 (f °g)(−1) = f [g(−1)] = −5

Figure 1.81

We can find a general expression for f 关 g共x兲兴 by evaluating f at g共x兲. f共x兲苷 2x 1 f 关 g共x兲兴苷 2关 g共x兲兴 1 苷 2关x 2 3兴 1 苷 2x 2 7

• Replace x by g (x). • Replace g(x) by x 2 3. • Simplify.

Thus f 关 g共x兲兴苷 2x 2 7. If we evaluate this function at 1, we have f [g共x兲] 苷 2x 2 7 f [g共1兲] 苷 2共1兲2 7 苷 2 7 苷 5 This is the same result that is obtained in Figure 1.81. In general, the composition of functions is not a commutative operation. That is, 共 f ⴰ g兲共x兲苷共 g ⴰ f 兲共x兲. To verify this, we will compute the composition 共 g ⴰ f 兲共x兲苷 g关 f共x兲兴, again using the functions f共x兲苷 2x 1 and g共x兲苷 x 2 3. g共x兲苷 x 2 3 g关 f共x兲兴苷关 f共x兲兴 2 3 苷关2x 1兴 2 3 苷 4x 2 4x 2

• Replace x by f (x). • Replace f (x) by 2x 1. • Simplify.

Thus g关 f共x兲兴苷 4x 2 4x 2, which is not equal to f 关 g共x兲兴苷 2x 2 7. Therefore, 共 f ⴰ g兲共x兲苷共 g ⴰ f 兲共x兲 and composition is not a commutative operation. QUESTION

Let f共x兲苷 x 1 and g共x兲苷 x 1. Then f 关 g共x兲兴苷 g关 f共x兲兴. (You should verify this statement.) Does this contradict the statement we made that composition is not a commutative operation?

EXAMPLE 5

»

Form Composite Functions

If f共x兲苷 x 2 3x and g共x兲苷 2x 1, find a.

共 g ⴰ f 兲共x兲

ANSWER

b.

共 f ⴰ g兲共x兲

No. When we say that composition is not a commutative operation, we mean that generally, given any two functions, 共 f ⴰ g兲共x兲苷共 g ⴰ f 兲共x兲. However, there may be particular instances in which 共 f ⴰ g兲共x兲苷共 g ⴰ f 兲共x兲. It turns out that these particular instances are quite important, as we shall see later.

1.5

The Algebra of Functions

75

Solution 共 g ⴰ f 兲共x兲苷 g关 f共x兲兴苷 2共 f共x兲兲 1 苷 2共x 2 3x兲 1 苷 2x 2 6x 1

a.

共 f ⴰ g兲共x兲苷 f 关 g共x兲兴苷共 g共x兲兲2 3共 g共x兲兲苷共2x 1兲2 3共2x 1兲苷 4x 2 2x 2

b.

• Substitute f (x) for x in g. • f (x) x 2 3x

• Substitute g(x) for x in f. • g(x) 2x 1

»

Try Exercise 38, page 78

Caution Some care must be used when forming the composition of functions. For instance, if f共x兲苷 x 1 and g共x兲苷兹x 4, then

共 g ⴰ f 兲共2兲苷 g关 f共2兲兴苷 g共3兲苷兹3 4 苷兹1 which is not a real number. We can avoid this problem by imposing suitable restrictions on the domain of f so that the range of f is part of the domain of g. If the domain of f is restricted to 关3, 兲, then the range of f is 关4, 兲. But this is precisely the domain of g. Note that 2 关3, 兲, and thus we avoid the problem of 共 g ⴰ f 兲共2兲 not being a real number. To evaluate 共 f ⴰ g兲共c兲 for some constant c, you can use either of the following methods. Method 1

First evaluate g共c兲. Then substitute this result for x in f共x兲.

Method 2

First determine f 关 g共x兲兴 and then substitute c for x.

EXAMPLE 6

»

Evaluate a Composite Function

Evaluate 共 f ⴰ g兲共3兲, where f共x兲苷 2x 7 and g共x兲苷 x 2 4.

Solution Method 1

take note In Example 6, both Method 1 and Method 2 produce the same result.

Method 2 共 f ⴰ g兲共x兲苷 2关 g共x兲兴 7

苷 2关x 4兴 7 苷 2x 2 1 共 f ⴰ g兲共3兲苷 2共3兲2 1 苷 19

• Evaluate g(3).

• Substitute 13 for x in f. • Form f [g(x)].

2

Although Method 2 is longer, it is the better method if you must evaluate 共f ⴰ g兲共x兲 for several values of x.

共 f ⴰ g兲共3兲苷 f 关 g共3兲兴苷 f 关共3兲2 4兴苷 f共13兲苷 2共13兲 7 苷 19

»

Try Exercise 50, page 78

• Substitute 3 for x.

76

Chapter 1

Functions and Graphs EXAMPLE 7

»

Use a Composite Function to Solve an Application

A graphic artist has drawn a 3-inch by 2-inch rectangle on a computer screen. The artist has been scaling the size of the rectangle for t seconds in such a way that the upper right corner of the original rectangle is moving to the right at the rate of 0.5 inch per second and downward at the rate of 0.2 inch per second. See Figure 1.82. t = 0 seconds

t = 3 seconds

t = 6 seconds

2 in. 3 in. t = 9 seconds

t = 12 seconds

Figure 1.82

a.

Write the length l and the width w of the scaled rectangles as functions of t.

b.

Write the area A of the scaled rectangle as a function of t.

c.

Find the intervals on which A is an increasing function for 0 t 14. Also find the intervals on which A is a decreasing function.

d.

Find the value of t (where 0 t 14) that maximizes A共t兲.

Solution a.

Because distance 苷 rate time, we see that the change in l is given by 0.5t. Therefore, the length at any time t is l 苷 3 0.5t. For 0 t 10, the width is given by w 苷 2 0.2t. For 10 t 14, the width is w 苷 2 0.2t. In either case the width can be determined by finding w 苷兩2 0.2t兩. (The absolute value symbol is needed to keep the width positive for 10 t 14.)

b.

A 苷 lw 苷共3 0.5t兲兩2 0.2t兩

c.

Use a graphing utility to determine that A is increasing on 关0, 2兴 and on 关10, 14兴 and that A is decreasing on 关2, 10兴. See Figure 1.83.

d.

The highest point on the graph of A occurs when t 苷 14 seconds. See

Area, in square inches

10 (2.0, 6.4)

(14, 8)

14

0 0 Time, in seconds

A 苷共3 0.5t兲兩2 0.2t兩

Figure 1.83

Figure 1.83.

»

Try Exercise 66, page 78

You may be inclined to think that if the area of a rectangle is decreasing, then its perimeter is also decreasing, but this is not always the case. For example, the area of the scaled rectangle in Example 7 was shown to decrease on 关2, 10兴 even though its perimeter is always increasing. See Exercise 68 in Exercise Set 1.5.

1.5

The Algebra of Functions

77

Topics for Discussion 1. The domain of f g consists of all real numbers formed by the union of the domain of f and the domain of g. Do you agree? 1 2 x , determine f ⴰ g and g ⴰ f. Do your 3 3 results show that composition of functions is a commutative operation?

2. Given f共x兲苷 3x 2 and g共x兲苷

3. A tutor states that the difference quotient of f共x兲苷 x 2 and the difference quotient of g共x兲苷 x 2 4 are the same. Do you agree? 4. A classmate states that the difference quotient of any linear function f共x兲苷 mx b is always m. Do you agree? 5. When we use a difference quotient to determine an average velocity, we generally replace the variable h with the variable t. What does t represent?

Exercise Set 1.5 In Exercises 1 to 12, use the given functions f and g to find f f g, f g, fg, and . State the domain of each. g

1. f 共x兲苷 x 2 2x 15, 2. f 共x兲苷 x 25, 2

3. f 共x兲苷 2x 8, 4. f 共x兲苷 5x 15,

g共x兲苷 x 4 g共x兲苷 x 3

6. f 共x兲苷 x 2 5x 8,

g共x兲苷 x g共x兲苷 x

7. f 共x兲苷 4x 7, g共x兲苷 2x 2 3x 5 8. f 共x兲苷 6x 10,

13. 共 f g兲共5兲

g共x兲苷 x 3

g共x兲苷 x 5

5. f 共x兲苷 x 3 2x 2 7x,

In Exercises 13 to 28, evaluate the indicated function, where f (x) x 2 3x 2 and g(x ) 2x 4.

冉冊

15. 共 f g兲

10. f 共x兲苷兹x 4, 11. f 共x兲苷兹4 x 2, 12. f 共x兲苷兹x 2 9,

g共x兲苷 x g共x兲苷 2 x g共x兲苷 x 3

2 3

18. 共 f g兲共24兲

19. 共 f g兲共1兲

20. 共 f g兲共0兲

21. 共 fg兲共7兲

22. 共 fg兲共3兲

冉冊冉冊冉冊冉冊

23. 共 fg兲

25.

g共x兲苷 3x 2 x 10 g共x兲苷 x

冉冊

16. 共 f g兲

17. 共 f g兲共3兲

27. 9. f 共x兲苷兹x 3,

1 2

14. 共 f g兲共7兲

2 5

24. 共 fg兲共100兲

f 共4兲 g

26.

f g

28.

1 2

冉冊冉冊冉冊 f 共11兲 g f g

1 4

In Exercises 29 to 36, find the difference quotient of the given function.

29. f 共x兲苷 2x 4

30. f 共x兲苷 4x 5

31. f 共x兲苷 x 2 6

32. f 共x兲苷 x 2 11

78

Chapter 1

Functions and Graphs

33. f 共x兲苷 2x 2 4x 3

34. f 共x兲苷 2x 2 5x 7

63. 共 g ⴰ h兲共k 1兲

35. f 共x兲苷 4x 2 6

36. f 共x兲苷 5x 2 4x

65. WATER TANK A water tank has the shape of a right circular cone with height 16 feet and radius 8 feet. Water is running into the tank so that the radius r (in feet) of the surface of the water is given by r 苷 1.5t, where t is the time (in minutes) that the water has been running.

In Exercises 37 to 48, find (g ⴰ f )(x ) and (f ⴰ g)(x ) for the given functions f and g.

37. f 共x兲苷 3x 5,

g共x兲苷 2x 7

38. f 共x兲苷 2x 7,

g共x兲苷 3x 2

39. f 共x兲苷 x 2 4x 1, 40. f 共x兲苷 x 2 11x,

g共x兲苷 x 2

16 ft

g共x兲苷 5x

42. f 共x兲苷 x 3 7,

g共x兲苷 x 1

2 , x1

44. f 共x兲苷兹x 4,

8 ft

g共x兲苷 2x 3

41. f 共x兲苷 x 3 2x,

43. f 共x兲苷

64. 共h ⴰ g兲共k 1兲

h

g共x兲苷 3x 5

g共x兲苷

a. The area A of the surface of the water is A 苷 p r 2. Find A共t兲 and use it to determine the area of the surface of the water when t 苷 2 minutes.

1 x

1 p r 2h. 3 Find V共t兲 and use it to determine the volume of the water when t 苷 3 minutes. (Hint: The height of the water in the cone is always twice the radius of the surface of the water.)

b. The volume V of the water is given by V 苷 45. f 共x兲苷

1 , x2

46. f 共x兲苷

6 , x2

g共x兲苷兹x 1

g共x兲苷

3 5x 66.

3 , 47. f 共x兲苷兩5 x兩 48. f 共x兲苷兩2x 1兩,

2 g共x兲苷 x g共x兲苷 3x 2 1

In Exercises 49 to 64, evaluate each composite function, where f (x) 2x 3, g(x) x 2 5x, and h(x) 4 3x 2.

49. 共 g ⴰ f 兲共4兲

50. 共 f ⴰ g兲共4兲

51. 共 f ⴰ g兲共3兲

52. 共 g ⴰ f 兲共1兲

53. 共 g ⴰ h兲共0兲

54. 共h ⴰ g兲共0兲

55. 共 f ⴰ f 兲共8兲

56. 共 f ⴰ f 兲共8兲

冉冊

57. 共h ⴰ g兲

2 5

SCALING A RECTANGLE Rework Example 7 of this section with the scaling as follows. The upper right corner of the original rectangle is pulled to the left at 0.5 inch per second and downward at 0.2 inch per second.

67. TOWING A BOAT A boat is towed by a rope that runs through a pulley that is 4 feet above the point where the rope is tied to the boat. The length (in feet) of the rope from the boat to the pulley is given by s 苷 48 t, where t is the time in seconds that the boat has been in tow. The horizontal distance from the pulley to the boat is d.

s d

冉冊

58. 共 g ⴰ h兲

1 3

59. 共 g ⴰ f 兲共兹3 兲

60. 共 f ⴰ g兲共兹2 兲

61. 共 g ⴰ f 兲共2c兲

62. 共 f ⴰ g兲共3k兲

a. Find d共t兲.

b. Evaluate s共35兲 and d共35兲.

1.5 PERIMETER OF A SCALED RECTANGLE Show by a graph that the perimeter

68.

The average rate of change of the concentration over the time interval from t 苷 a to t 苷 a t is C共a t兲 C共a兲 t

P 苷 2共3 0.5t兲 2兩2 0.2t兩 of the scaled rectangle in Example 7 of this section is an increasing function over 0 t 14. 69. CONVERSION FUNCTIONS The function F共x兲苷

x con12

x converts 3 x feet to yards. Explain the meaning of 共Y ⴰ F兲共x兲. verts x inches to feet. The function Y共x兲苷

70. CONVERSION FUNCTIONS The function F共x兲苷 3x converts x yards to feet. The function I共x兲苷 12x converts x feet to inches. Explain the meaning of 共I ⴰ F兲共x兲. 71.

CONCENTRATION OF A MEDICATION The concentration C共t兲 (in milligrams per liter) of a medication in a patient’s blood is given by the data in the following table.

Concentration of Medication in Patient’s Blood C(t) C(t) mg/l

0

0

0.25

47.3

0.50

78.1

0.75

94.9

1.00

99.8

1.25

95.7

1.50

84.4

1.75

68.4

2.00

50.1

2.25

31.6

2.50

15.6

2.75

4.3

100 Concentration (in mg/liter)

t hours

80

Use the data in the table to evaluate the average rate of change for each of the following time intervals. a. 关0, 1兴 (Hint: In this case, a 苷 0 and t 苷 1.) Compare this result to the slope of the line through 共0, C共0兲兲 and 共1, C共1兲兲. b. 关0, 0.5兴

c. 关1, 2兴

d. 关1, 1.5兴

e. 关1, 1.25兴

f. The data in the table can be modeled by the function Con共t兲苷 25t 3 150t 2 225t. Use Con共t兲 to verify that the average rate of change over 关1, 1 t兴 is 75共t兲 25共t兲2. What does the average rate of change over 关1, 1 t兴 seem to approach as t approaches 0? 72. BALL ROLLING ON A RAMP The distance traveled by a ball rolling down a ramp is given by s共t兲苷 6t 2, where t is the time in seconds after the ball is released, and s共t兲 is measured in feet. The ball travels 6 feet in 1 second and 24 feet in 2 seconds. Use the difference quotient for average velocity given on page 71 to evaluate the average velocity for each of the following time intervals. a. 关2, 3兴 (Hint: In this case, a 苷 2 and t 苷 1.) Compare this result to the slope of the line through 共2, s 共2兲兲 and 共3, s 共3兲兲.

60

b. 关2, 2.5兴

40

c. 关2, 2.1兴

d. 关2, 2.01兴

e. 关2, 2.001兴

f. Verify that the average velocity over 关2, 2 t兴 is 24 6共t兲. What does the average velocity seem to approach as t approaches 0?

20 0

79

The Algebra of Functions

0

1

2

3

t

Time (in hours)

Connecting Concepts In Exercises 73 to 76, show that (f ⴰ g)(x) (g ⴰ f )(x).

73. f 共x兲苷 2x 3;

g共x兲苷 5x 12

74. f 共x兲苷 4x 2;

g共x兲苷 7x 4

5x x2

75. f 共x兲苷

6x ; x1

g共x兲苷

76. f 共x兲苷

5x ; x3

g共x兲苷

2x x4

80

Chapter 1

Functions and Graphs

In Exercises 77 to 82, show that (g ⴰ f )(x ) x

and

(f ⴰ g)(x ) x

79. f 共x兲苷

4 , x1

g共x兲苷

4x x

2 , 1x

g共x兲苷

x2 x

77. f 共x兲苷 2x 3,

g共x兲苷

x3 2

80. f 共x兲苷

78. f 共x兲苷 4x 5,

g共x兲苷

x5 4

81. f 共x兲苷 x 3 1, 82. f 共x兲苷 x 3 2,

3

g共x兲苷兹 x 1 3

g共x兲苷兹 2 x

Projects 1.

Now consider the function

A GRAPHING UTILITY PROJECT For any two different real numbers x and y, the larger of the two numbers is given by Maximum共x, y兲苷

x y 兩x y兩 2 2

y3 苷共 y1 y2 兲兾2 共abs共 y1 y2 兲兲兾2 where “abs” represents the absolute value function. The graph of y3 is shown below.

(1)

8

a. Verify Equation (1) for x 苷 5 and y 苷 9. b. Verify Equation (1) for x 苷 201 and y 苷 80. For any two different function values f 共x兲 and g共x兲, the larger of the two is given by Maximum共 f 共x兲, g共x兲兲苷

f 共x兲 g共x兲兩 f 共x兲 g共x兲兩 2 2

−1

(2)

6

y3 苷共 y1 y2 兲兾2 共abs 共 y1 y2 兲兲兾2

To illustrate how we might make use of Equation (2), consider the functions y1 苷 x 2 and y2 苷兹x on the interval from Xmin 苷 1 to Xmax 苷 6. The graphs of y1 and y2 are shown below. 8

c.

Write a sentence or two that explains why the graph of y3 is as shown.

d.

What is the domain of y1? of y2? of y3? Write a sentence that explains how to determine the domain of y3, given the domain of y1 and the domain of y2.

e. Determine a formula for the function Minimum共 f 共x兲, g共x兲兲 −1

6 1

y1 苷 x 2

2.

THE NEVER-NEGATIVE FUNCTION The author J. D. Murray describes a function f that is defined in the following manner.1

8

f 苷

再

f, 0,

if f 0 if f 0

We will refer to this function as a never-negative function. Never-negative functions can be graphed by using Equation (2) in Project 1. For example, if we let g共x兲苷 0, then Equation (2) simplifies to −1

6

Maximum共 f 共x兲, 0兲苷

1

y2 苷兹x 1Mathematical

f 共x兲兩 f 共x兲兩 2 2

Biology (New York: Springer-Verlag, 1989), p. 101.

(3)

1.6 The graph of y 苷 Maximum共 f 共x兲, 0兲 is the graph of y 苷 f 共x兲 provided that f 共x兲 0, and it is the graph of y 苷 0 provided that f 共x兲 0.

An Application The mosquito population per acre of a large resort is controlled by spraying on a monthly basis. A biologist has determined that the mosquito population can be approximated by the never-negative function M with M共t兲苷 35,400共t int共t兲兲2 35,400共t int共t兲兲 4000 Here t represents the month, and t 苷 0 corresponds to June 1, 2004.

Section 1.6 ■

■

■

■

Introduction to Inverse Functions Graphs of Inverse Functions Composition of a Function and Its Inverse Find an Inverse Function

Inverse Functions

81

a. Use a graphing utility to graph M for 0 t 3. b. Use a graphing utility to graph M for 0 t 3. Write a sentence or two that explains how the graph of M differs from the graph of M.

c.

d. What is the maximum mosquito population per acre for 0 t 3? When does this maximum mosquito population occur? e.

Explain when would be the best time to visit the resort, provided that you wished to minimize your exposure to mosquitos.

Inverse Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A6.

PS1. Solve 2x 5y 苷 15 for y. [1.1] PS2. Solve x 苷

y1 for y. [1.1] y

PS3. Given f共x兲苷

2x 2 , find f共1兲. [1.3] x1

PS4. If f is a function and f共3兲苷 7, write an ordered pair of the function. [1.3] PS5. Let f共x兲苷 3x 7. What is the domain of f? [1.3] PS6. Let f共x兲苷兹x 2. What is the domain of f? [1.3]

Introduction to Inverse Functions Consider the “doubling function” f共x兲苷 2x that doubles every input. Some of the ordered pairs of this function are

再

共4, 8兲, 共1.5, 3兲, 共1, 2兲,

冉冊

冎

5 10 , , 共7, 14兲 3 3

1 x that takes one-half of every 2 input. Some of the ordered pairs of this function are Now consider the “halving function” g共x兲苷

再

共8, 4兲, 共3, 1.5兲, 共2, 1兲,

冉冊

冎

10 5 , , 共14, 7兲 3 3

82

Chapter 1

Functions and Graphs Observe that the ordered pairs of g are the ordered pairs of f with the order of the coordinates reversed. The following two examples illustrate this concept.

take note In this section our primary interest concerns finding the inverse of a function; however, we can also find

f共5兲苷 2共5兲苷 10 Ordered pair: (5, 10)

g共10兲苷

1 共10兲苷 5 2

Ordered pair: (10, 5)

the inverse of a relation. Recall that a relation r is any set of ordered pairs. The inverse of r is the set of ordered pairs formed

f共a兲苷 2共a兲苷 2a Ordered pair: (a, 2a)

by reversing the order of the

g共2a兲苷

1 共2a兲苷 a 2

Ordered pair: (2a, a)

coordinates of the ordered pairs in r.

The function g is said to be the inverse function of f. Definition of an Inverse Function If the ordered pairs of a function g are the ordered pairs of a function f with the order of the coordinates reversed, then g is the inverse function of f.

Consider a function f and its inverse function g. Because the ordered pairs of g are the ordered pairs of f with the order of the coordinates reversed, the domain of the inverse function g is the range of f, and the range of g is the domain of f. Not all functions have an inverse that is a function. Consider, for instance, the “square function” S共x兲苷 x 2. Some of the ordered pairs of S are y 10 (−3, 9)

兵共3, 9兲, 共1, 1兲, 共0, 0兲, 共1, 1兲, 共3, 9兲, 共5, 25兲其 If we reverse the coordinates of the ordered pairs, we have

(3, 9)

8

兵共9, 3兲, 共1, 1兲, 共0, 0兲, 共1, 1兲, 共9, 3兲, 共25, 5兲其

6 S(x) = x2

4 2

−4

−2

2

Figure 1.84

4

x

This set of ordered pairs is not a function because there are ordered pairs, for instance 共9, 3兲 and 共9, 3兲, with the same first coordinate and different second coordinates. In this case, S has an inverse relation but not an inverse function. A graph of S is shown in Figure 1.84. Note that x 苷 3 and x 苷 3 produce the same value of y. Thus the graph of S fails the horizontal line test, and therefore S is not a one-to-one function. This observation is used in the following theorem. Condition for an Inverse Function A function f has an inverse function if and only if f is a one-to-one function.

Recall that increasing functions and decreasing functions are one-to-one functions. Thus we can state the following theorem. Alternative Condition for an Inverse Function If f is an increasing or a decreasing function, then f has an inverse function.

1.6 QUESTION

83

Inverse Functions

Which of the functions graphed below has an inverse function?

y

y

y

g

f

h x

x

x

take note f 1共x兲 does not mean f 共x兲苷 2x, f 1共x兲苷

1 . For f 共x兲

If a function g is the inverse of a function f, we usually denote the inverse function by f 1 rather than g. For the doubling and halving functions f and g discussed on page 81, we write

1 x but 2

1 1 苷 . f 共x兲 2x

f 1共x兲苷

f共x兲苷 2x

1 x 2

Graphs of Inverse Functions Because the coordinates of the ordered pairs of the inverse of a function f are the ordered pairs of f with the order of the coordinates reversed, we can use them to create a graph of f 1. y

EXAMPLE 1

f 6 4 2 (−1, 0.5) −4

−2

Solution

(1, 2) (0, 1)

−2

Sketch the Graph of the Inverse of a Function

Sketch the graph of f 1 given that f is the function shown in Figure 1.85.

(2, 4)

2

»

4

6

x

Because the graph of f passes through 共1, 0.5兲, 共0, 1兲, 共1, 2兲, and 共2, 4兲, the graph of f 1 must pass through 共0.5, 1兲, 共1, 0兲, 共2, 1兲, and 共4, 2兲. Plot the points and then draw a smooth curve through the points, as shown in Figure 1.86.

−4 y f

y=x

6

Figure 1.85

4 (1, 2) (0, 1) 2 (−1, 0.5) −4

−2 −2

(2, 4)

f −1

(2, 1) 2 4 (1, 0) (0.5, −1)

(4, 2) 6

x

−4

Figure 1.86

»

Try Exercise 10, page 91

ANSWER

The graph of f is the graph of an increasing function. Therefore, f is a one-toone function and has an inverse function. The graph of h is the graph of a decreasing function. Therefore, h is a one-to-one function and has an inverse function. The graph of g is not the graph of a one-to-one function. g does not have an inverse function.

84

Chapter 1

Functions and Graphs The graph from the solution to Example 1 is shown again in Figure 1.87. Note that the graph of f 1 is symmetric to the graph of f with respect to the graph of y 苷 x. If the graph were folded along the dashed line, the graph of f would lie on top of the graph of f 1. This is a characteristic of all graphs of functions and their inverses. In Figure 1.88, although S does not have an inverse that is a function, the graph of the inverse relation S 1 is symmetric to S with respect to the graph of y 苷 x. y f

y=x

6 4 (1, 2) (0, 1) 2 (−1, 0.5) −4

−2 −2

(2, 4)

y

f −1

y=x

4 S

(2, 1) 2 4 (1, 0) (0.5, −1)

2

(4, 2) 6

x

−4

−2

2

4

x

−2

−4

−4

Figure 1.87

S −1

Figure 1.88

Composition of a Function and Its Inverse Observe the effect of forming the composition of f (x) 苷 2x and g(x) 苷 f共x兲苷 2x

冋册

f 关 g共x兲兴苷 2

take note If we think of a function as a machine, then the Composition of Inverse Functions Property can be represented as shown below. Take any input x for f. Use the output of

f as the input for f 1. The result is the original input, x. x

1 x 2

1 x 2 1 g关 f共x兲兴苷关2x兴 2 g关 f共x兲兴苷 x

1 x. 2

g共x兲苷

• Replace x by g(x).

f 关 g共x兲兴苷 x

• Replace x by f (x).

This property of the composition of inverse functions always holds true. When taking the composition of inverse functions, the inverse function reverses the effect of the original function. For the two functions above, f doubles a number, and g halves a number. If you double a number and then take one-half of the result, you are back to the original number.

Composition of Inverse Functions Property If f is a one-to-one function, then f 1 is the inverse function of f if and only if

f (x) f function x f −1 function

共 f ⴰ f 1 兲共x兲苷 f 关 f 1共x兲兴苷 x

for all x in the domain of f 1

共 f 1 ⴰ f 兲共x兲苷 f 1关 f共x兲兴苷 x

for all x in the domain of f.

and

1.6 EXAMPLE 2

»

85

Inverse Functions

Use the Composition of Inverse Functions Property

Use composition of functions to show that f 1共x兲苷 3x 6 is the inverse 1 function of f共x兲苷 x 2. 3

Solution We must show that f 关 f 1共x兲兴苷 x and f 1关 f共x兲兴苷 x. f共x兲苷 f 关 f 1共x兲兴苷

1 x2 3

f 1共x兲苷 3x 6

1 关3x 6兴 2 3

f 关 f 1共x兲兴苷 x

f 1关 f共x兲兴苷 3

冋

册

1 x2 6 3

f 1关 f共x兲兴苷 x

»

Try Exercise 20, page 91

Integrating Technology In the standard viewing window of a calculator, the distance between two tick marks on the x-axis is not equal to the distance between two tick marks on the y-axis. As a result, the graph of y 苷 x does not appear to bisect the first and third quadrants. See Figure 1.89. This anomaly is important if a graphing calculator is being used to check whether two functions are inverses of one another. Because the graph of y 苷 x does not appear to bisect the first and third quadrants, the graphs of f and f 1 will not appear to be symmetric 1 about the graph of y 苷 x. The graphs of f共x兲苷 x 2 and f 1共x兲苷 3x 6 3 from Example 2 are shown in Figure 1.90. Notice that the graphs do not appear to be quite symmetric about the graph of y 苷 x.

10

10

−10

Distances between tick marks are not equal.

10

−10

10

−10

−10

y = x in the standard viewing window

f , f −1, and y = x in the standard viewing window

Figure 1.89

Figure 1.90 Continued

䉴

86

Chapter 1

Functions and Graphs To get a better view of a function and its inverse, it is necessary to use the SQUARE viewing window, as in Figure 1.91. In this window, the distance between two tick marks on the x-axis is equal to the distance between two tick marks on the y-axis.

10 Distances are equal. −15

15

−10

f , f −1, and y = x in a square viewing window

Figure 1.91

Find an Inverse Function If a one-to-one function f is defined by an equation, then we can use the following method to find the equation for f 1.

take note If the ordered pairs of f are given by (x, y), then the ordered pairs of f 1

Steps for Finding the Inverse of a Function To find the equation of the inverse f 1 of the one-to-one function f :

are given by ( y, x). That is, x and y

1. Substitute y for f共x兲.

are interchanged. This is the reason

2. Interchange x and y.

for Step 2 at the right.

3. Solve, if possible, for y in terms of x. 4. Substitute f 1共x兲 for y.

EXAMPLE 3

»

Find the Inverse of a Function

Find the inverse of f共x兲苷 3x 8.

Solution f共x兲苷 3x 8 y 苷 3x 8

• Replace f (x) by y.

x 苷 3y 8

• Interchange x and y.

x 8 苷 3y

• Solve for y.

1.6

Inverse Functions

87

x8 苷y 3 1 8 x 苷 f 1共x兲 3 3

• Replace y by f 1.

The inverse function is given by f 1共x兲苷

1 8 x . 3 3

»

Try Exercise 32, page 91

In the next example we find the inverse of a rational function. EXAMPLE 4

»

Find the Inverse of a Function

Find the inverse of f共x兲苷

2x 1 , x 苷 0. x

Solution f共x兲苷

2x 1 x

y苷

2x 1 x

• Replace f (x) by y.

x苷

2y 1 y

• Interchange x and y.

xy 苷 2y 1

• Solve for y.

xy 2y 苷 1 y共x 2兲苷 1 y苷 f 1共x兲苷

• Factor the left side.

1 x2 1 ,x苷2 x2

• Replace y by f 1.

»

Try Exercise 38, page 92

QUESTION

If f is a one-to-one function and f共4兲苷 5, what is f 1共5兲?

The graph of f共x兲苷 x 2 4x 3 is shown in Figure 1.92a. The function f is not a one-to-one function and therefore does not have an inverse function. However,

ANSWER

Because (4, 5) is an ordered pair of f, (5, 4) must be an ordered pair of f 1. Therefore, f 1共5兲苷 4.

88

Chapter 1

Functions and Graphs the function given by G共x兲苷 x 2 4x 3, shown in Figure 1.92b, for which the domain is restricted to 兵x 兩 x 2其, is a one-to-one function and has an inverse function G 1. This is shown in Example 5.

y

f

−4

y

4

4

2

2

−2

2

4

x

−2

2

−2

−2

−4

−4

Figure 1.92a

EXAMPLE 5

−4

»

G

4

x

Figure 1.92b

Find the Inverse of a Function with a Restricted Domain

Find the inverse of G共x兲苷 x 2 4x 3, where the domain of G is 兵x 兩 x 2其.

Solution

take note

G共x兲苷 x 2 4x 3 y 苷 x 2 4x 3 x 苷 y 2 4y 3 x 苷共 y 2 4y 4兲 4 3

Recall that the range of a function

f is the domain of f 1, and the 1

domain of f is the range of f .

y

−4

x 苷共 y 2兲2 1 x 1 苷共 y 2兲2

G

4

y=x

2

G −1

−2

2 −2 −4

Figure 1.93

4

兹x 1 苷兹共 y 2兲2

兹x 1 苷 y 2

x

• Replace G(x) by y. • Interchange x and y. • Solve for y by completing the square of y 2 4y. • Factor. • Add 1 to each side of the equation. • Take the square root of each side of the equation. • Recall that if a 2 b, then a 兹b.

兹x 1 2 苷 y Because the domain of G is 兵x 兩 x 2其, the range of G 1 is 兵 y 兩 y 2其. This means that we must choose the positive value of 兹x 1. Thus G 1共x兲苷兹x 1 2. See Figure 1.93.

»

Try Exercise 44, page 92

In Example 6b, we use an inverse function to determine the wholesale price of a gold bracelet for which we know the retail price.

1.6 EXAMPLE 6

»

Inverse Functions

Solve an Application

A merchant uses the function S共x兲苷

4 x 100 3

to determine the retail selling price S, in dollars, of a gold bracelet for which she has paid a wholesale price of x dollars. a.

The merchant paid a wholesale price of $672 for a gold bracelet. Use S to determine the retail selling price of this bracelet.

b.

Find S 1 and use it to determine the merchant’s wholesale price for a gold bracelet that retails at $1596.

Solution 4 共672兲 100 苷 896 100 苷 996 3 The merchant charges $996 for a bracelet that has a wholesale price of $672.

a.

S共672兲苷

b.

To find S 1, begin by substituting y for S共x兲. 4 3 4 y苷 3 4 x苷 3 4 x 100 苷 3 S共x兲苷

x 100 x 100

• Replace S(x ) with y.

y 100

• Interchange x and y.

y

• Solve for y.

3 共x 100兲苷 y 4 3 x 75 苷 y 4 Using inverse notation, the above equation can be written as S 1共x兲苷

3 x 75 4

Substitute 1596 for x to determine the wholesale price. 3 共1596兲 75 4 苷 1197 75 苷 1122

S 1共1596兲苷

A gold bracelet that the merchant retails at $1596 has a wholesale price of $1122.

»

Try Exercise 54, page 92

89

90

Chapter 1

Functions and Graphs

Integrating Technology Some graphing utilities can be used to draw the graph of the inverse of a function without the user having to find the inverse function. For instance, Figure 1.94 shows the graph of f共x兲苷 0.1x 3 4. The graphs of f and f 1 are both shown in Figure 1.95, along with the graph of y 苷 x. Note that the graph of f 1 is the reflection of the graph of f with respect to the graph of y 苷 x. The display shown in Figure 1.95 was produced on a TI-83/TI-83 Plus/TI-84 Plus graphing calculator by using the DrawInv command, which is in the DRAW menu. 10

10

y=x

f −1 − 15

15

−15

15

3

3

f(x) = 0.1x − 4

f(x) = 0.1x − 4

−10

−10

Figure 1.94

Figure 1.95

Topics for Discussion 1. If f共x兲苷 3x 1, what are the values of f 1共2兲 and 关 f共2兲兴 1? 2. How are the domain and range of a one-to-one function f related to the domain and range of the inverse function of f ? 3. How is the graph of the inverse of a function f related to the graph of f ? 4. The function f共x兲苷 x is its own inverse. Find two other functions that are their own inverses. 5. What are the steps in finding the inverse of a one-to-one function?

Exercise Set 1.6 In Exercises 1 to 4, assume that the given function has an inverse function.

1. Given f 共3兲苷 7, find f 1共7兲. 2. Given g共3兲苷 5, find g 1共5兲. 3. Given h1共3兲苷 4, find h共4兲. 4. Given f 1共7兲苷 0, find f 共0兲.

5. If 3 is in the domain of f 1, find f 关 f 1共3兲兴. 6. If f is a one-to-one function and f 共0兲苷 5, f 共1兲苷 2, and f 共2兲苷 7, find: a. f 1共5兲

b. f 1共2兲

7. The domain of the inverse function f 1 is the of f. 8. The range of the inverse function f 1 is the

of f.

1.6 In Exercises 9 to 16, draw the graph of the inverse relation. Is the inverse relation a function? y

9.

y

10.

8

8 (8, 6)

4

4

(2, 3)

(− 4, 0) −8 − 4 (−8, −2) − 4

4

8

x

−8

−4

4 −4

8

17. f 共x兲苷 4x; g共x兲苷

x 4

18. f 共x兲苷 3x; g共x兲苷

1 3x

x

19. f 共x兲苷 4x 1; g共x兲苷

(6, −3)

20. f 共x兲苷 y 8

8

(−6, 4) 4

4

−4 −4

x 8 (6, −5)

−8

(−3, 0)

(0, 3)

−4

4

−8

1 1 x ; g共x兲苷 2x 1 2 2

22. f 共x兲苷 3x 2; g共x兲苷 8

x

−4

(−5, −6)

(0, −3)

−8

2 1 x 3 3

23. f 共x兲苷

5 5 ; g共x兲苷 3 x3 x

24. f 共x兲苷

2x x ; g共x兲苷 x1 x2 3

y

25. f 共x兲苷 x 3 2; g共x兲苷兹 x 2

8

8

26. f 共x兲苷共x 5兲3; g共x兲苷兹 x 5

4

4

y

13.

21. f 共x兲苷 (2, 6)

4

−8

1 3 x ; g共x兲苷 2x 3 2 2

y

12.

(−4, 0) −8

1 1 x 4 4

−8

−8

11.

91

In Exercises 17 to 26, use composition of functions to determine whether f and g are inverses of one another.

(0, 6)

(2, 3)

Inverse Functions

14.

3

−4

−4

In Exercises 27 to 30, find the inverse of the function. If the function does not have an inverse function, write “no inverse function.”

−8

−8

27. 兵共3, 1兲, 共2, 2兲, 共1, 5兲, 共4, 7兲其

−4

4

8

x

−8

−4

4

8

x

28. 兵共5, 4兲, 共2, 3兲, 共0, 1兲, 共3, 2兲, 共7, 11兲其 29. 兵共0, 1兲, 共1, 2兲, 共2, 4兲, 共3, 8兲, 共4, 16兲其 y

15.

−8

y

16.

8

8

4

4

−4

4

8

x

−8

−4

30. 兵共1, 0兲, 共10, 1兲, 共100, 2兲, 共1000, 3兲, 共10,000, 4兲其

4

−4

−4

−8

−8

8

x

In Exercises 31 to 48, find f 1(x). State any restrictions on the domain of f 1(x).

31. f 共x兲苷 2x 4 32. f 共x兲苷 4x 8 33. f 共x兲苷 3x 7 34. f 共x兲苷 3x 8

92

Chapter 1

Functions and Graphs

35. f 共x兲苷 2x 5

to determine the retail selling price S, in dollars, of a winter coat for which she has paid a wholesale price of x dollars.

36. f 共x兲苷 x 3 2x 37. f 共x兲苷 , x1

x苷1

a. The merchant paid a wholesale price of $96 for the winter coat. Use S to determine the retail selling price she will charge for this coat.

38. f 共x兲苷

x , x2

x苷2

b. Find S 1 and use it to determine the merchant’s wholesale price for a coat that retails at $399.

39. f 共x兲苷

x1 , x1

x 苷 1

40. f 共x兲苷

2x 1 , x3

x0

42. f 共x兲苷 x 2 4,

x0

43. f 共x兲苷兹x 2 ,

x2

44. f 共x兲苷兹4 x ,

x4

45. f 共x兲苷 x 2 4x ,

x 2

46. f 共x兲苷 x 2 6x ,

x3

47. f 共x兲苷 x 2 4x 1 ,

x 2

48. f 共x兲苷 x 2 6x 1 ,

x3

50.

FASHION The function s共x兲苷 2x 24 can be used to convert a U.S. women’s shoe size into an Italian women’s shoe size. Determine the function s1共x兲 that can be used to convert an Italian women’s shoe size to its equivalent U.S. shoe size.

54.

FASHION The function K共x兲苷 1.3x 4.7 converts a men’s shoe size in the United States to the equivalent shoe size in the United Kingdom. Determine the function K 1共x兲 that can be used to convert a United Kingdom men’s shoe size to its equivalent U.S. shoe size.

55.

COMPENSATION The monthly earnings E共s兲, in dollars, of a software sales executive is given by E共s兲苷 0.05s 2500, where s is the value, in dollars, of the software sold by the executive during the month. Find E 1共s兲 and explain how the executive could use this function.

56.

POSTAGE Does the first-class postage rate function given below have an inverse function? Explain your answer.

x 苷 3

41. f 共x兲苷 x 2 1,

49.

53.

GEOMETRY The volume of a cube is given by V共x兲苷 x 3, where x is the measure of the length of a side of the cube. Find V 1共x兲 and explain what it represents. UNIT CONVERSIONS The function f 共x兲苷 12x converts feet, x, into inches, f 共x兲. Find f 1共x兲 and explain what it determines.

51. FAHRENHEIT TO CELSIUS The function f 共x兲苷

5 共x 32兲 9

is used to convert x degrees Fahrenheit to an equivalent Celsius temperature. Find f 1 and explain how it is used. 52. RETAIL SALES A clothing merchant uses the function S共x兲苷

3 x 18 2

Weight (in ounces)

Cost

0w1

$.39

1w2

$.63

2w3

$.87

3w4

$1.11

57. THE BIRTHDAY PROBLEM A famous problem called the birthday problem goes like this: Suppose there is a randomly selected group of n people in a room. What is the probability that at least two of the people have a birthday on the same day of the year? It may surprise you that for a group of 23 people, the probability that at least two of the people share a birthday is about 50.7%. The following graph can be used to estimate shared birthday probabilities for 1 n 60.

1.6

59. CRYPTOLOGY Cryptology is the study of making and breaking secret codes. Secret codes are often used to send messages over the Internet. By devising a code that is difficult to break, the sender hopes to prevent the messages from being read by an unauthorized person. In practice, very complicated one-to-one functions and their inverses are used to encode and decode messages. The following procedure uses the simple function f 共x兲苷 2x 1 to illustrate the basic concepts that are involved. Assign to each letter of the alphabet, and a blank space, a two-digit numerical value, as shown below.

p

Probability that at least two people in the group share the same birthday

1.0

p(n)

0.9 0.8 0.7 0.6 0.5 0.4 0.3

A

0.2 0.1 0

10

20

30

40

50

60

n

Number of people in the group

a. Use the graph of p to estimate p共10兲 and p共30兲. b. Consider the function p with 1 n 60, as shown in the graph. Explain how you can tell that p has an inverse that is a function. c.

Write a sentence that explains the meaning of p1共0.223兲 in the context of this application.

Milligrams of pseudoephedrine hydrochloride in the bloodstream

L(t) = 0.03t 4 + 0.4t 3 − 7.3t 2 + 23.1t

10

H 17

O

24

V

B

11

I

C

12

J

D

13

E

14

F

15

G

16

18

P

25

W 32

19

Q

26

X

33

K

20

R

27

Y

34

L

21

S

28

Z

35

M 22

T

29

N 23

U

30

36

Using these numerical values, the message MEET YOU AT NOON would be represented by 22 14 14 29 36 34 24 30 36 10 29 36 23 24 24 23 Let f 共x兲苷 2x 1 define a coding function. The above message can be encoded by finding f 共22兲, f 共14兲, f 共14兲, f 共29兲, f 共36兲, f 共34兲, f 共24兲, … , f 共23兲, which yields 43 27 27 57 71 67 47 59 71 19 57 71 45 47 47 45 The inverse of f , which is f 1共x兲苷

x1 2

is used by the receiver of the message to decode the message. For instance,

20

f 1共43兲苷

16

43 1 苷 22 2

which represents M, and

12

f 1共27兲苷

8 4

0

31

Note: A blank space is represented by the numerical value 36.

58. MEDICATION LEVEL The function L shown in the following graph models the level of pseudoephedrine hydrochloride, in milligrams, in the bloodstream of a patient t hours after 30 milligrams of the medication have been administered. L

93

Inverse Functions

27 1 苷 14 2

which represents E. 1

2 3 4 Time (in hours)

t

a. Use the graph of L to estimate two different values of t for which the pseudoephedrine hydrochloride levels are the same.

a. Use the above coding procedure to encode the message DO YOUR HOMEWORK. b. Use f 1共x兲 as defined above to decode the message 49 33 47 45 27 71 33 47 43 27. c.

b. Does L have an inverse that is a function? Explain.

Explain why it is important to use a one-to-one function to encode a message.

94

Chapter 1

Functions and Graphs

60. CRYPTOGRAPHY A friend is using the letter-number correspondence in Exercise 59 and the coding function g共x兲苷 2x 3. Your friend sends you the coded message 59 31 39 73 31 75 61 37 31 75 29 23 71 Use g 1共x兲 to decode this message. In Exercises 61 to 66, answer the question without finding the equation of the linear function.

61.

Suppose that f is a linear function, f 共2兲苷 7, and f 共5兲苷 12. If f 共4兲苷 c, then is c less than 7, between 7 and 12, or greater than 12? Explain your answer.

62.

Suppose that f is a linear function, f 共1兲苷 13, and f 共4兲苷 9. If f 共3兲苷 c, then is c less than 9, between 9 and 13, or greater than 13? Explain your answer.

63. Suppose that f is a linear function, f 共2兲苷 3, and f 共5兲苷 9. Between which two numbers is f 1共6兲? 64. Suppose that f is a linear function, f 共5兲苷 1, and f 共9兲苷 3. Between which two numbers is f 1共2兲? 65. Suppose that g is a linear function, g 1共3兲苷 4, and g 1共7兲苷 8. Between which two numbers is g共5兲? 66. Suppose that g is a linear function, g 1共2兲苷 5, and g 1共0兲苷 3. Between which two numbers is g共0兲?

Connecting Concepts 67. Consider the linear function f 共x兲苷 mx b, m 苷 0. The graph of f has a slope of m and a y-intercept of 共0, b兲. What are the slope and y-intercept of the graph of f 1? 68. Find the inverse of f 共x兲苷 ax 2 bx c, a 苷 0, x

b . 2a

69. Use a graph of f 共x兲苷 x 3 to explain why f is its own inverse. 70. Use a graph of f 共x兲苷兹16 x 2 , with 0 x 4, to explain why f is its own inverse.

Only one-to-one functions have inverses that are functions. In Exercises 71 to 74, determine if the given function is a oneto-one function.

71. p共t兲苷兹 9 t 72. v共t兲苷兹 16 t 73. F共x兲苷兩x兩 x 74. T共x兲苷兩x 2 6兩,

x0

Projects INTERSECTION POINTS FOR THE GRAPHS OF f AND f 1. For each of the following, graph f and its inverse.

1. i.

f 共x兲苷 2x 4

ii. f 共x兲苷 x 2 iii. f 共x兲苷 x 3 1 iv. f 共x兲苷 x 3 v. f 共x兲苷 3x 2 vi. f 共x兲苷

1 x

a. Do the graphs of a function and its inverse always intersect? b. If the graphs of a function and its inverse intersect at one point, what is true about the coordinates of the point of intersection? c. Can the graphs of a function and its inverse intersect at more than one point?

1.7

Modeling Data Using Regression

Section 1.7 ■ ■ ■

95

Modeling Data Using Regression

Linear Regression Models Correlation Coefficient Quadratic Regression Models

P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A6.

PS1. Find the slope and y-intercept of the graph of y 苷

3 x 4. 4

PS2. Find the slope and y-intercept of the graph of 3x 4y 苷 12. PS3. Find an equation of the line that has slope 0.45 and y-intercept 共0, 2.3兲. PS4. If f 共x兲苷 3x2 4x 1, find f 共2兲. [1.3] PS5. Find the distance between P1共3, 5兲 and P2共3, f 共3兲兲, where f共x兲苷 2x 5. [1.2/1.3] PS6. You are given P1共2, 1兲 and P2共4, 14兲. If f 共x兲苷 x2 3, find 兩 f 共x1兲 y1兩兩 f 共x2兲 y2兩. [1.3]

Linear Regression Models The data in the table below show the population of selected states and the number of professional sports teams (Major League Baseball, National Football League, National Basketball Association, Women’s National Basketball Association, National Hockey League) in those states. A scatter diagram of the data is shown in Figure 1.96 on page 96.

Number of Professional Sports Teams for Selected States State

Arizona

Populations (in millions)

Number of Teams

5.9

5

California

36.1

17

Colorado

4.7

4

Florida

17.8

11

Illinois

12.8

5

6.3 10.1

Indiana Michigan

State

Minnesota New Jersey

Populations (in millions)

5.1

Number of Teams

5

8.7

3

19.3

10

9.7

3

Pennsylvania

12.4

7

3

Texas

22.9

9

5

Wisconsin

5.5

3

New York North Carolina

96

Chapter 1

Functions and Graphs Although there is no one line that passes through every point, we could find an approximate linear model for these data. For instance, the line shown in Figure 1.97 in blue approximates the data better than the line shown in red. However, as Figure 1.98 shows, there are many other lines we could draw that seem to approximate the data.

10 5

5

10

15

20

25

30

35

P

15 10 5

0

Population (in millions)

Number of teams

15

0

T 20

T 20 Number of teams

Number of teams

T 20

5

10

15

20

25

30

35

P

15 10 5

0

5

Population (in millions)

Figure 1.96

Figure 1.97

10

15

20

25

30

35

P

Population (in millions)

Figure 1.98

To find the line that “best” approximates the data, regression analysis is used. This type of analysis produces the linear function whose graph is called the line of best fit or the least-squares regression line.2

Definition of the Least-Squares Regression Line The least-squares regression line is the line that minimizes the sum of the squares of the vertical deviations of all data points from the line.

To help understand this definition, consider the data set S 苷兵共1, 2兲, 共2, 3兲, 共3, 3兲,共4, 4兲, 共5, 7兲其 as shown in Figure 1.99. As we will show later, the least-squares line for this data set is y 苷 1.1x 0.5, also shown in Figure 1.99. If we evaluate this linear function at the x-coordinates of the data set S, we obtain the set of ordered pairs T 苷兵共1, 1.6兲, 共2, 2.7兲, 共3, 3.8兲,共4, 4.9兲, 共5, 6兲}. The vertical deviations are the differences between the y-coordinates in S and the y-coordinates in T. From the definition, we must calculate the sum of the squares of these deviations. 共2 1.6兲2 共3 2.7兲2 共3 3.8兲2 共4 4.9兲2 共7 6兲2 苷 2.7 Because y 苷 1.1x 0.5 is the least squares regression line, for no other line is the sum of the squares of the deviations less than 2.7. For instance, if we consider the equation y 苷 1.25x 0.75, which is the equation of the line through the two

2The

least-squares regression line is also called the least-squares line and the regression line.

1.7

97

Modeling Data Using Regression

points P1共1, 2兲 and P2共5, 7兲 of the data set, the sum of the squared deviations is larger than 2.7. See Figure 1.100. 共2 2兲2 共3 3.25兲2 共3 4.5兲2 共4 5.75兲2 共7 7兲2 苷 5.375 y

y

7

0.0

7 1.0

6 5

6 4

−0.8

3

3

0.3

2

− 0.25

2

0.4

1

−1.75

5

−0.9

4

−1.5

0.0

1 1

2

3

4

5

6

7

x

1

Figure 1.99

2

3

4

5

6

7

x

Figure 1.100

Integrating Technology The equations used to calculate a regression line are somewhat cumbersome. Fortunately, these equations are preprogrammed into most graphing calculators. We will now illustrate the technique for a TI-83/TI-83 Plus/TI-84 Plus calculator using data set S given on page 96.

Press

STAT

Press

ENTER

. Select EDIT. .

EDIT CALC TESTS 1 : Edit... 2 : SortA( 3 : SortD( 4 : ClrList 5 : SetUpEditor

Press STAT . Select CALC. Select 4, LinReg(ax+b).

If necessary, delete any data in L1 and L2. Enter the given data. L1

L2

1 2 3 4 5

2 3 3 4 7

Press

L3

2

L2(6) =

ENTER

.

Press

VARS

.

EDIT CALC TESTS 1 : 1–Var Stats 2 : 1–Var Stats 3 : Med–Med 4 : LinReg(ax+b) 5 : QuadReg 6 : CubicReg 7 QuartReg

LinReg(ax+b)

ENTER

View the results.

Select Y-VARS. Press

ENTER

.

VARS Y–VARS 1 : Function... 2 : Parametric... 3 : Polar... 4 : On/Off...

Select 1. Press FUNCTION 1 : Y1 2 : Y2 3 : Y3 4 : Y4 5 : Y5 6 : Y6 7 Y7

ENTER

.

Press

.

LinReg(ax+b) Y1

LinReg y=ax+b a=1.1 b=.5 r2=.8175675676 r=.9041944302

Continued

䉴

98

Chapter 1

Functions and Graphs From the last screen, the equation of the regression line is y 苷 1.1x 0.5. Your last screen may not look exactly like ours. The information provided on our screen requires that DiagnosticOn be enabled. This is accomplished using the following keystrokes: 2ND CATALOG (Scroll to DiagnosticOn) ENTER ENTER With DiagnosticOn enabled, in addition to the values for the regression equation, two other values, r 2 and r, are given. We will discuss these values later in this section. If you used the keystrokes we have shown here, the regression line will be stored in Y1. This is helpful if you wish to graph the regression line. However, if it is not necessary to graph the regression line, then instead of pressing VARS at step 4, just press ENTER . The result will be the last screen showing the results of the regression calculations.

EXAMPLE 1

»

Find a Regression Equation

Find the regression equation for the data on the population of a state and the number of professional sports teams in that state. How many sports teams are predicted for Indiana, whose population is approximately 6.3 million? Round to the nearest whole number.

Solution Using your calculator, enter the data from the table on page 95. Then have the calculator produce the values for the regression equation. Your results should be similar to those shown in Figure 1.101. The equation of the regression line is

Integrating Technology If you followed the steps we gave on page 97 and stored the regression equation from Example 1 in Y1, then you can evaluate the regression equation using the following keystrokes: VARS 䉴 ENTER ENTER ( 6.3 )

ENTER

LinReg y=ax+b a=.4285944776 b=1.000728509 r2=.8725655709 r= . 9 3 4 1 1 2 1 8 3 3

y 苷 0.4285944776x 1.000728509

Figure 1.101

To find the number of sports teams the regression equation predicts for Indiana, evaluate the regression equation for x 苷 6.3. y 苷 0.4285944776x 1.000728509 苷 0.4285944776(6.3兲 1.000728509 ⬇ 3.7008737 The equation predicts that Indiana should have four sports teams.

»

Try Exercise 18, page 104

1.7

99

Modeling Data Using Regression

Correlation Coefficient 20

0

40 0

Figure 1.102

The scatter diagram of a state’s population and the corresponding number of professional sports teams is shown in Figure 1.102, along with the graph of the regression line. Note that the slope of the regression line is positive. This indicates that as a state’s population increases, the number of teams increases. Note also that for these data the value of r on the regression calculation screen was positive, r ⬇ 0.9341. Now consider the data in the table below, which shows the trade-in value of a 2003 Corvette for various odometer readings. The graph in Figure 1.103 shows the scatter diagram and the regression line. Trade-in Value of 2003 Corvette Coupe, January 2006

take note The data for the Corvette were

Odometer Reading, in thousands of miles

Trade-in Value in dollars

condition of the car was excellent.

25

29,100

The only variable that changed

30

28,000

was the odometer reading.

40

27,175

45

26,450

55

24,225

created assuming that the

30,000

60

20 20,000

Figure 1.103

Source: Kelley Blue Book website, January 2006.

LinReg y=ax+b a= - 1 5 0 . 7 4 5 6 1 4 b=32869.07895 r2=.9635182628 r= - .9815896611

In this case the slope of the regression line is negative. This means that as the odometer reading increases, the trade-in value of the car decreases. Note also that the value of r is negative, r ⬇ 0.9816. See Figure 1.104. Linear Correlation Coefficient

Figure 1.104

The linear correlation coefficient r is a measure of how close the points of a data set can be modeled by a straight line. If r 苷 1, then the points of the data set can be modeled exactly by a straight line with negative slope. If r 苷 1, then the data set can be modeled exactly by a straight line with positive slope. For all data sets, 1 r 1.

y

y

r = 1.

y

y

x

x r is close to 1.

x r is close to 0.

y

r is close to −1.

x

r = −1.

x

If r 苷 1 or r 苷 1, then the data set cannot be modeled exactly by a straight line. The further the value of r is from 1 or 1 (the closer the value of r to zero), the more the ordered pairs of the data set deviate from a straight line.

100

Chapter 1

Functions and Graphs The graphs below show the points of the data sets and the graphs of the regression lines for the state population/sports teams data and the odometer reading/trade-in data. Note the values of r and the closeness of the data points to the regression lines. 30,000

20

0

40 0

r 苷 0.9341

60

20 20,000

r 苷 0.9816

A researcher calculates a regression line to determine the relationship between two variables. The researcher wants to know whether a change in one variable produces a predictable change in the second variable. The value of r 2 tells the researcher the extent of that relationship.

Coefficient of Determination The coefficient of determination r 2 measures the proportion of the variation in the dependent variable that is explained by the regression line.

For the population/sports team data, r 2 ⬇ 0.87. This means that approximately 87% of the total variation in the dependent variable (number of teams) can be attributed to the state population. This also means that population alone does not predict with certainty the number of sports teams. Other factors, such as climate, are also involved in the number of sports teams. QUESTION

y

What is the coefficient of determination for the odometer reading/ trade-in value data (see page 99), and what is its significance?

Quadratic Regression Models To this point our focus has been linear regression equations. However, there may be a nonlinear relationship between two quantities. The scatter diagram to the left suggests that a quadratic function might be a better model of the data than a linear model. x Nonlinear correlation between x and y.

ANSWER

r 2 ⬇ 0.964. This means that about 96.4% of the total variation in trade-in value can be attributed to the odometer reading.

1.7 EXAMPLE 2

»

Modeling Data Using Regression

101

Find a Quadratic Regression Model

The data in the table below were collected on five successive Saturdays. They show the average number of cars entering a shopping center parking lot. The value of t is the number of minutes after 9:00 A.M. The value of N is the number of cars that entered the parking lot in the 10 minutes prior to the value of t. Find a regression model for this data. Average Number of Cars Entering a Shopping Center Parking Lot t

N

t

N

20

70

140

301

40

135

160

298

60

178

180

284

80

210

200

286

100

260

220

260

120

280

240

195

Solution 1. Construct a scatter diagram for these data. Enter the data into your calculator as explained on page 97.

400

0

300 50

From the scatter diagram, it appears that there is a nonlinear relationship between the variables. 2. Find the regression equation. Try a quadratic regression model. For a TI-83/TI-83 Plus/T1-84 Plus calculator, press STAT 䉴 5 ENTER .

take note For nonlinear regression

QuadReg y=ax2+bx+c a=-.0124881369 b=3.904433067 c=-7.25 R2=.9840995401

calculations, the value of r is not

shown on a TI-83/T1-83 Plus/T1-84 Plus graphing calculator. In such cases, the coefficient of determination is used to determine how well the data fit the model.

3. Examine the coefficient of determination. The coefficient of determination is approximately 0.984. Because this number is fairly close to 1, the regression equation y 苷 0.0124881369x 2 3.904433067x 7.25 provides a good model of the data. Continued

䉴

102

Chapter 1

Functions and Graphs The following calculator screen shows the scatter diagram and the parabola that is the graph of the regression equation from page 101. 400

0

300 50

»

Try Exercise 32, page 107

LinReg y=ax+b a=.6575174825 b=144.2727273 r2=.4193509866 r=.6475731515

In Example 2, we could have calculated the linear regression model for the data. The results are shown at the left. Note that the coefficient of determination for this calculation is approximately 0.419. Because this number is less than the coefficient of determination for the quadratic model, we choose a quadratic model of the data rather than a linear model. A final note: The regression line equation does not prove that the changes in the dependent variable are caused by the independent variable. For instance, suppose various cities throughout the United States were randomly selected and the numbers of gas stations (independent variable) and restaurants (dependent variable) were recorded in a table. If we calculated the regression equation for these data, we would find that r would be close to 1. However, this does not mean that gas stations cause restaurants to be built. The primary cause is that there are fewer gas stations and restaurants in cities with small populations and greater numbers of gas stations and restaurants in cities with large populations.

Topics for Discussion 1. What is the purpose of calculating the equation of a regression line? 2. Discuss the implications of the following correlation coefficients: r 苷 1, r 苷 0, and r 苷 1. 3. Discuss the coefficient of determination and what its value says about a data set. 4. What are the implications of r 2 苷 1 for a quadratic regression equation?

1.7

103

Modeling Data Using Regression

Exercise Set 1.7 In Exercises 7 to 12, find the linear regression equation for the given set.

Use a graphing calculator for this Exercise Set. In Exercises 1 to 4, determine whether the scatter diagram suggests a linear relationship between x and y, a nonlinear relationship between x and y, or no relationship between x and y.

1.

y

2.

7. 兵共2, 6兲, 共3, 6兲, 共4, 8兲, 共6, 11兲, 共8, 18兲其 8. 兵共2, 3兲, 共3, 4兲, 共4, 9兲, 共5, 10兲, 共7, 12兲其 9. 兵共3, 11.8兲, 共1, 9.5兲, 共0, 8.6兲, 共2, 8.7兲, 共5, 5.4兲其

y

10. 兵共7, 11.7兲, 共5, 9.8兲, 共3, 8.1兲, 共1, 5.9兲, 共2, 5.7兲其 11. 兵共1.3, 4.1兲, 共2.6, 0.9兲, 共5.4, 1.2兲, 共6.2, 7.6兲, 共7.5, 10.5兲其 12. 兵共1.5, 8.1兲, 共0.5, 6.2兲, 共3.0, 2.3兲, 共5.4, 7.1兲, 共6.1, 9.6兲其 x

x

3.

y

4.

y

In Exercises 13 to 16, find a quadratic model for the given data.

13. 兵共1, 1兲, 共2, 1兲, 共4, 8兲, 共5, 14兲, 共6, 25兲其 14. 兵共2, 5兲, 共1, 0兲, 共0, 1兲, 共1, 4兲, 共2, 4兲其 x

x

15. 兵共1.5, 2.2兲, 共2.2, 4.8兲, 共3.4, 11.2兲, 共5.1, 20.6兲, 共6.3, 28.7兲其 16. 兵共2, 1兲, 共1, 3.1兲, 共0, 2.9兲, 共1, 0.8兲, 共2, 6.8兲, 共3, 15.9兲其

In Exercises 5 and 6, determine for which scatter diagram, A or B, the coefficient of determination is closer to 1.

5.

y

y

17.

ARCHEOLOGY The data below show the length of the humerus and the wingspan, in centimeters, of several pterosaurs, which are extinct flying reptiles of the order Pterosauria. (Source: Southwest Educational Development Laboratory.)

Pterosaur Data

x

x

Figure A 6.

Figure B

y

y

x

Figure A

x

Figure B

Humerus (cm)

Wingspan (cm)

Humerus (cm)

Wingspan (cm)

24

600

20

500

32

750

27

570

22

430

15

300

17

370

15

310

13

270

9

240

4.4

68

4.4

55

3.2

53

2.9

50

1.5

24

a. Compute the linear regression equation for these data.

104

Chapter 1

Functions and Graphs with pure water having a pH of 7) on the growth of a bacteria culture. The table below gives the measurements of different cultures, in thousands of bacteria, after 8 hours.

b. On the basis of this model, what is the projected wingspan of the pterosaur Quetzalcoatlus northropi, which is thought to have been the largest of the prehistoric birds, if its humerus is 54 centimeters? Round to the nearest centimeter.

Number of Bacteria (in thousands)

pH

18.

SPORTS The data in the table below show the distance, in feet, a ball travels for various swing speeds, in miles per hour, of a bat.

120

6

131

Distance (ft)

7

136

40

200

8

141

45

213

9

151

50

242

10

148

60

275

11

163

70

297

75

326

80

335

b. Using the regression model, what is the expected distance a ball will travel when the bat speed is 58 miles per hour? Round to the nearest foot.

a. Find the linear regression equation for these data. b. Using the regression model, what is the expected number of bacteria when the pH is 7.5? Round to the nearest thousand bacteria.

SPORTS The table below shows the number of strokes per minute that a rower makes and the speed of the boat in meters per second.

21.

HEALTH The body mass index (BMI) of a person is a measure of the person’s ideal body weight. The table below shows the BMI for different weights for a person 5 feet 6 inches tall. (Source: San Diego UnionTribune, May 31, 2000)

BMI Data for Person 5 6 Tall Weight (lb)

BMI

Weight (lb)

BMI

Strokes per minute

Speed (m/s)

110

17

160

25

30

4.1

120

19

170

27

31

4.2

125

20

180

29

33

4.4

135

21

190

30

34

4.5

140

22

200

32

36

4.7

145

23

205

33

39

5.1

150

24

215

34

a. Find the linear regression equation for these data.

a. Compute the linear regression equation for these data.

b. Using the regression model, what is the expected speed of the boat when the rowing rate is 32 strokes per minute? Round to the nearest tenth of a meter per second. 20.

116

5

Bat speed (mph)

a. Find the linear regression equation for these data.

19.

4

BIOLOGY A medical researcher wanted to determine the effect of pH (a measure of alkalinity or acidity,

b. On the basis of the model, what is the estimated BMI for a person 5 feet 6 inches tall whose weight is 158 pounds? 22.

HEALTH The BMI (see Exercise 21) of a person depends on height as well as weight. The table

1.7 below shows the changes in BMI for a 150-pound person as height (in inches) changes. (Source: San Diego UnionTribune, May 31, 2000)

Average Remaining Lifetime for Men Age

BMI Data for 150-Pound Person Height (in.)

BMI

Height (in.)

BMI

60

29

71

21

62

27

72

20

64

25

73

19

66

24

74

19

67

23

75

18

68

23

76

18

70

21

Years

Age

Years

0

73.6

65

15.9

15

59.4

75

9.9

35

40.8

Based on the data in this table, is there a strong correlation between a man’s age and the average remaining lifetime for that man? Explain. 25.

HEALTH SCIENCES The average remaining lifetime for women in the United States is given in the following table. (Source: National Institutes of Health.) Average Remaining Lifetime for Women Age

a. Compute the linear regression equation for these data. b. On the basis of the model, what is the estimated BMI for a 150-pound person who is 5 feet 8 inches tall? 23. INDUSTRIAL ENGINEERING Permanent-magnet direct-current motors are used in a variety of industrial applications. For these motors to be effective, there must be a strong linear relationship between the current (in amps, A) supplied to the motor and the resulting torque (in newton-centimeters, N-cm) produced by the motor. A randomly selected motor is chosen from a production line and tested, with the following results.

a.

Years

Age

Years

0

79.4

65

19.2

15

65.1

75

12.1

35

45.7

Based on the data in this table, is there a strong correlation between a woman’s age and the average remaining lifetime for that woman?

b. Compute the linear regression equation for these data. c. On the basis of the model, what is the estimated remaining lifetime of a woman of age 25? 26.

Direct-Current Motor Data at 12 Volts

BIOLOGY The table below gives the body lengths, in centimeters, and the highest observed flying speeds, in meters per second, of various animals.

Current, in A

Torque, in N-cm

Current, in A

Torque, in N-cm

7.3

9.4

8.5

8.6

11.9

2.8

7.9

4.3

Horsefly

1.3

6.6

5.6

5.6

14.5

9.5

Hummingbird

8.1

11.2

14.2

4.9

12.7

8.3

Dragonfly

7.9

7.0

10.6

4.7

Willow warbler

11

12.0

Common pintail

56

22.8

Based on the data in this table, is the chosen motor effective? Explain. 24.

105

Modeling Data Using Regression

HEALTH SCIENCES The average remaining lifetime for men in the United States is given in the following table. (Source: National Institutes of Health.)

Species

Length (cm)

8.5

Flying speed (m/s)

10.0

Based on these data, what is the flying speed of a Whimbrel whose length is 41 centimeters? Round to the nearest meter per second. (Source: Based on data from Leiva, Algebra 2: Explorations and Applications, p. 76, McDougall Littell, Boston; copyright 1997.)

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Chapter 1

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27. HEALTH The table below shows the number of calories burned in 1 hour when running at various speeds. Running Speed (mph)

Calories Burned

10

1126

10.9

1267

5

563

5.2

633

6

704

6.7

774

7

809

8

950

8.6

985

9

1056

7.5

880

Larvae Surviving for Various Temperatures Temp. (C)

Number Surviving

Temp. (C)

Number Surviving

20

40

26

68

21

47

27

67

22

52

28

64

23

61

29

62

24

64

30

61

25

64

30. METEOROLOGY The temperature at various times on a summer day at a resort in southern California is given in the following table. The variable t is the number of minutes after 6:00 A.M., and the variable T is the temperature in degrees Fahrenheit. Temperatures at a Resort t (min)

T (F )

t (min)

T (F )

a. Are the data positively or negatively correlated?

20

59

240

86

b. How many calories does this model predict a person who runs at 9.5 mph for 1 hour will burn? Round to the nearest calorie.

40

65

280

88

80

71

320

86

120

78

360

85

160

81

400

80

200

83

28. TRAFFIC SAFETY A traffic safety institute measured the braking distance, in feet, of a car traveling at certain speeds in miles per hour. The data from one of those tests are given in the following table.

a. Find a quadratic model for these data.

Speed (mph)

Breaking Distance (ft)

20

23.9

30

33.7

40

40.0

50

41.7

60

46.8

70

48.9

mph

mpg

mph

mpg

80

49.0

25

29

55

31

30

32

60

28

35

33

65

24

40

35

70

19

45

34

75

17

50

33

a. Find the quadratic regression equation for these data. b. Using the regression model, what is the expected braking distance when a car is traveling at 65 mph? Round to the nearest tenth of a foot. 29. BIOLOGY The survival of certain larvae after hatching depends on the temperature (in degrees Celsius) of the surrounding environment. The following table shows the number of larvae that survive at various temperatures. Find a quadratic model for these data.

b. Use the model to predict the temperature at 1:00 P.M. Round to the nearest tenth of a degree. 31. AUTOMOTIVE ENGINEERING The fuel efficiency, in miles per gallon, for a certain midsize car at various speeds, in miles per hour, is given in the table below. Fuel Efficiency of a Midsize Car

a. Find a quadratic model for these data. b. Use the model to predict the fuel efficiency of this car when it is traveling at a speed of 50 mph.

1.7

107

Modeling Data Using Regression

32. BIOLOGY The data in the table at the right show the oxygen consumption, in milliliters per minute, of a bird flying level at various speeds in kilometers per hour.

Oxygen Consumption Speed (km/h)

Consumption (ml/min)

20

32

25

27

28

22

35

21

42

26

50

34

a. Find a quadratic model for these data. b. Use the model to predict the oxygen consumption of a bird flying at 40 kilometers per hour. Round to the nearest milliliter per minute.

Connecting Concepts 33. PHYSICS Galileo (1564 – 1642) studied acceleration due to gravity by allowing balls of various weights to roll down an incline. This allowed him to time the descent of a ball more accurately than by just dropping the ball. The data in the table show some possible results of such an experiment using balls of different masses. Time t is measured in seconds; distance s is measured in centimeters. Distance Traveled for Balls of Various Weights 5-Pound Ball t s

10-Pound Ball t s

15-Pound Ball t s

Science, 168). His paper dealt with the distance an extragalactic nebula was from the Milky Way galaxy and the nebula’s velocity with respect to the Milky Way. The data are given in the table below. Distance is measured in megaparsecs (1 megaparsec equals 1.918 1019 miles), and velocity (called the recession velocity) is measured in kilometers per second. A negative velocity means the nebula is moving toward the Milky Way; a positive velocity means the nebula is moving away from the Milky Way. Recession Velocities

2

2

3

5

3

5

Distance

4

10

6

22

5

15

0.032

170

0.9

650

6

22

9

49

7

30

0.034

290

0.9

150

8

39

12

87

9

49

0.214

130

0.9

500

10

61

15

137

11

75

0.263

70

1.0

920

12

86

18

197

13

103

0.275

185

1.1

450

14

120

15

137

0.275

220

1.1

500

16

156

0.45

200

1.4

500

0.5

290

1.7

960

0.5

270

2.0

500

0.63

200

2.0

850

0.8

300

2.0

800

0.9

30

2.0

1090

a. Find a quadratic model for each of the balls. b.

On the basis of a similar experiment, Galileo concluded that if air resistance is excluded, all falling objects fall with the same acceleration. Explain how one could make such a conclusion from the regression equations.

Velocity

Distance

Velocity

a. Find the linear regression model for these data. 34.

ASTRONOMY In 1929, Edwin Hubble published a paper that revolutionized astronomy (“A Relationship Between Distance and Radial Velocity Among ExtraGalactic Nebulae,” Proceedings of the National Academy of

b. On the basis of this model, what is the recession velocity of a nebula that is 1.5 megaparsecs from the Milky Way?

108

Chapter 1

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Projects MEDIAN–MEDIAN LINE Another linear model of data is called the median–median line. This line employs summary points calculated using the medians of subsets of the independent and dependent variables. The median of a data set is the middle number or the average of the two middle numbers for a data set arranged in numerical order. For instance, to find the median of 兵8, 12, 6, 7, 9其, first arrange the data in numerical order.

x

y

2

3

3

5

4

4

5

7

6

8

6, 7, 8, 9, 12

7

9

The median is 8, the number in the middle. To find the median of 兵15, 12, 20, 9, 13, 10其, arrange the numbers in numerical order.

8

12

9

12

10

14

11

15

12

14

9, 10, 12, 13, 15, 20 The median is 12.5, the average of the two middle numbers. 12 13 Median 苷苷 12.5 2 The median – median line is determined by dividing a data set into three equal groups. (If the set cannot be divided into three equal groups, the first and third groups should be equal. For instance, if there are 11 data points, divide the set into groups of 4, 3, and 4.) The slope of the median – median line is the slope of the line through the x-medians and y-medians of the first and third sets of points. The median – median line passes through the average of the x-and y-medians of all three sets. A graphing calculator can be used to find the equation of the median – median line. This line, along with the linear regression line, is shown in the next column for the data in the accompanying table. 1. Find the equation of the median – median line for the data in Exercise 17 on page 103. 2. Find the equation of the median – median line for the data in Exercise 18 on page 104.

16

Regression line

0

14 0

Median-median line

3. Consider the data set 兵共1, 3兲, 共2, 5兲, 共3, 7兲, 共4, 9兲, 共5, 11兲, 共6, 13兲, 共7, 15兲, 共8, 17兲其. a. Find the equation of the linear regression line for these data. b. Find the equation of the median–median line for these data. c.

What conclusion might you draw from the answers to a. and b.?

4. For this exercise, use the data in the table above. a. Calculate the equation of the median–median line and the linear regression line. b. Change the entry 共12, 14兲 to 共12, 1兲 and then recalculate the equations of the median–median line and the linear regression line. c.

Explain why there is more change in the linear regression line than in the median–median line.

Exploring Concepts with Technology Graphing Piecewise Functions with a Graphing Calculator A graphing calculator can be used to graph piecewise functions by including as part of the function the interval on which each piece of the function is defined. The method is based on the fact that a graphing calculator “evaluates” inequalities. For purposes of this Exploration, we will use keystrokes for a TI-83/TI-83 Plus/TI-84 Plus calculator.

109

Exploring Concepts with Technology

For instance, store 3 in X by pressing 3 STO䉴 X,T,,n ENTER . Now enter the inequality x 4 by pressing X,T,,n 2ND TEST 3 4 ENTER . Your screen

3 ––> X 3

should look like the one at the left. Note that the value of the inequality on the screen is 0. This occurs because the calculator replaced X by 3 and then determined whether the inequality 3 4 was true or false. The calculator expresses the fact that the inequality is false by placing a zero on the screen. If we repeat the sequence of steps above, except that we store 5 in X instead of 3, the calculator will determine that the inequality is true and place a 1 on the screen. This property of calculators is used to graph piecewise functions. Graphs of these functions work best when Dot mode rather than Connected mode is used. To switch to Dot mode, select MODE , use the arrow keys to highlight DOT , and then press ENTER .

X>4 0

Next we will graph the piecewise function defined by 10

f共x兲苷 −10

10

−10

再

x, x 2,

x 2 x 2

Enter the function3 as Y1=X*(X≤-2)+X2*(X>-2) and graph it in the standard viewing window. Note that you are multiplying each piece of the function by its domain. The graph will appear as shown at the left. To understand how the graph is drawn, we will consider two values of x, 8 and 2, and evaluate Y1 for each of these values. Y1=X*(X≤-2)+X2*(X>-2) 苷 8共8 2兲共8兲2共8 2兲苷 8共1兲 64共0兲苷 8 • When x 8, the value assigned to 8 2 is 1; the value assigned to 8 2 is 0.

(X≤-2)+X2

Y1=X* *(X>-2) 苷 2共2 2兲 22共2 2兲苷 2共0兲 4共1兲苷 4

●

When x 2, the value assigned to 2 2 is 0; the value assigned to 2 2 is 1.

In a similar manner, for any value of x, x 2, the value assigned to (X≤–2) is 1 and the value assigned to (X>–2) is 0. Thus Y1=X*1+X2*0=X on that interval. This means that only the f共x兲苷 x piece of the function is graphed. When x 2, the value assigned to (X≤–2) is 0 and the value assigned to (X>–2) is 1. Thus Y1=X*0+X2*1=X2 on that interval. This means that only the f共x兲苷 x 2 piece of the function is graphed on that interval. 1. Graph: f共x兲苷 3. Graph: f共x兲苷

3Note

再再

x 2, x,

x2 x2

x 2 1, x 2 1,

x0 x0

2.

Graph: f共x兲苷

4.

Graph: f共x兲苷

that pressing 2ND TEST will display the inequality menu.

再再

x 2 x, x 4,

x2 x2

x 3 4x, x x 2, 2

x1 x1

110

Chapter 1

Functions and Graphs

Chapter 1 Summary 1.1 Equations and Inequalities

• If f is a function and c is a positive constant, then the graph of

• The integers are the set 兵. . . , 3, 2, 1, 0, 1, 2, 3, . . .其. The positive integers are called natural numbers. The rational a numbers are a, b integers and b 苷 0 . Irrational numbers b are nonterminating and nonrepeating decimals. The real numbers are the set of rational and irrational numbers.

再兩

冎

• The interval of real numbers 共a, b兲苷兵x 兩 a x b其. • A linear equation is one that can be written in the form ax b 苷 0, a 苷 0. A quadratic equation is one that can be written in the form ax 2 bx c 苷 0, a 苷 0. • An inequality is a statement that contains the symbol , , , or .

1.2 A Two-Dimensional Coordinate System and Graphs • The Distance Formula The distance d between the points represented by 共x1 , y1 兲 and 共x2 , y2 兲 is d 苷兹共x2 x1 兲共 y2 y1 兲

2

2

• The midpoint of the line segment from P1 共x1 , y1 兲 to P2 共x2 , y2 兲 is

冉

冊

x1 x2 y1 y2 , 2 2

• The standard form of the equation of a circle with center at 共h, k兲 and radius r is 共x h兲2 共 y k兲2 苷 r 2.

1.3 Introduction to Functions • Definition of a Function A function is a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates. • A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point. If every horizontal line intersects the graph of a function at most once, then the graph is the graph of a one-to-one function.

1.4 Properties of Graphs • The graph of an equation is symmetric with respect to the y-axis if the replacement of x with x leaves the equation unaltered. the x-axis if the replacement of y with y leaves the equation unaltered. the origin if the replacement of x with x and y with y leaves the equation unaltered.

y 苷 f 共x兲 c is the graph of y 苷 f 共x兲 shifted up vertically c units y 苷 f 共x兲 c is the graph of y 苷 f 共x兲 shifted down vertically c units y 苷 f 共x c兲 is the graph of y 苷 f 共x兲 shifted left horizontally c units y 苷 f 共x c兲 is the graph of y 苷 f 共x兲 shifted right horizontally c units • The graph of y 苷 f 共x兲 is the graph of y 苷 f 共x兲 reflected across the x-axis. y 苷 f 共x兲 is the graph of y 苷 f 共x兲 reflected across the y-axis. • If c 1, then the graph of y 苷 c f 共x兲 is a vertical stretching of the graph of y 苷 f共x兲. • If 0 c 1, then the graph of y 苷 c f 共x兲 is a vertical compression of the graph of y 苷 f共x兲. • If c 1, then the graph of y 苷 f 共cx兲 is a horizontal compression of the graph of y 苷 f共x兲. • If 0 c 1, then the graph of y 苷 f 共cx兲 is a horizontal stretching of the graph of y 苷 f共x兲.

1.5 The Algebra of Functions • For all values of x for which both f 共x兲 and g共x兲 are defined, we define the following functions. Sum

共 f g兲共x兲苷 f 共x兲 g共x兲

Difference

共 f g兲共x兲苷 f 共x兲 g共x兲

Product

共 f g兲共x兲苷 f 共x兲 g共x兲

Quotient

冉冊

f 共x兲 f 共x兲苷 , g g共x兲

• The expression

g共x兲苷 0

f 共x h兲 f 共x兲 , h

h苷0

is called the difference quotient of f. The difference quotient can be used to compute the average velocity of a moving object over a given time interval. • For the functions f and g, the composition of the two functions is given by 共 f ⴰ g兲共x兲苷 f 关 g共x兲兴 for all x in the domain of g such that g共x兲 is in the domain of f.

1.6 Inverse Functions • If f is a one-to-one function with domain X and range Y, and g is a function with domain Y and range X, then g is the inverse function of f if and only if 共 f ⴰ g兲共x兲苷 x for all x in the domain of g and 共 g ⴰ f 兲共x兲苷 x for all x in the domain of f.

Assessing Concepts • A function f has an inverse function if and only if it is a oneto-one function. The graph of a function f and the graph of its inverse function f 1 are symmetric with respect to the line given by y 苷 x.

1.7 Modeling Data Using Regression

111

• The linear correlation coefficient r is a measure of how closely the points of a data set can be modeled by a straight line. If r 苷 1, then the points of the data set can be modeled exactly by a straight line with negative slope. If r 苷 1, then the data set can be modeled exactly by a straight line with positive slope. For all data sets, 1 r 1.

• Regression analysis is used to find a mathematical model of collected data.

• The coefficient of determination is r 2. It measures the percent of the total variation in the dependent variable that is explained by the regression line.

• The least-squares regression line is the line that minimizes the sum of the squares of the vertical deviations of all data points from the line.

• It is possible to find both linear and nonlinear mathematical models of data.

Chapter 1 Assessing Concepts 1. A function f has the property that for two real numbers a and b, with a b, f 共a兲 f 共b兲. Which of the following are true statements? a. f is a one-to-one function. b. f is an increasing function. c. f is a decreasing function. d. f has an inverse function. e. f could be a linear function. f. f could be a quadratic function. 2. Let f共x兲苷 3x 8 and g共x兲苷 2x 4. Show that f 关g共x兲] 苷 g关 f 共x兲兴 Are f and g inverse functions? 3. If f共x兲苷 f共x 4兲 for all real numbers x, and f共2兲苷 3, find f共18兲. 4. Use absolute value notation to describe the set of points that are less than 3 units from the point whose coordinate is 2 on the real number line. 5. If a and b are two different numbers in the domain of a f(b) f(a) function f, explain the meaning of . ba

6. Suppose f is a one-to-one function and f共2兲苷 3. Find the coordinates of an ordered pair on the graph of the inverse of f. 7. Suppose 共2, 3兲 are the coordinates of a point on the graph of y 苷 f共x兲. Give the coordinates of a point on the graph of y 苷 f共x 5兲. 8. Suppose 共3, 5兲 are the coordinates of a point on the graph of y 苷 f共x兲. Give the coordinates of a point on the graph of y 苷 f共x兲 1. 9. Suppose 共6, 4兲 are the coordinates of a point on the graph of y 苷 f共x兲. Give the coordinates of a point on the graph of y 苷 f共2x兲. 10. Suppose that the correlation coefficient for a linear regression equation is 0.86. Does this information enable you to determine anything about the slope of the regression line?

112

Chapter 1

Functions and Graphs

Chapter 1 Review Exercises In Exercises 1 to 14, solve each equation or inequality.

1. 3 4z 苷 12

In Exercises 25 to 32, determine whether the graph of each equation is symmetric with respect to the x-axis, the y-axis, or the origin.

2. 4y 3 苷 6y 5

3. 2x 3共2 3x兲苷 14x

25. y 苷 x 2 7

26. x 苷 y 2 3

4. 5 2共3m 2兲苷 3共1 m兲

27. y 苷 x 3 4x

28. y 2 苷 x 2 4

5. y 2 3y 18 苷 0

6. 2z 2 9z 4 苷 0

7. 3v v 苷 1

8. 3s 苷 4 2s

2

29.

2

30. xy 苷 8

31. 兩 y兩苷兩x兩

9. 3c 5 5c 7 10. 7a 5 2共3a 4兲 11. x 2 x 12 0

12. 2x 2 x 1

13. 兩2x 5兩 3

14. 兩1 3x兩 4

33. 共x 3兲2 共 y 4兲2 苷 81 34. x 2 y 2 10x 4y 20 苷 0

15. a 苷 6, b 苷 8. Find c.

16. a 苷 11, c 苷 20. Find b.

17. b 苷 12, c 苷 13. Find a.

18. a 苷 7, b 苷 14. Find c.

20. 共5, 4兲共3, 8兲

22. 共4, 7兲共8, 11兲

a. f 共1兲

b. f 共3兲

c. f 共t兲

d. f 共x h兲

e. 3f 共t兲

f. f 共3t兲

a. g共3兲

b. g共5兲

c. g共8兲

d. g共x兲

e. 2g共t兲

f. g共2t兲

a. 共 f ⴰ g兲共3兲

b. 共g ⴰ f 兲共3兲

c. 共 f ⴰ g兲共x兲

d. 共g ⴰ f 兲共x兲

40. If f共x兲苷 2x 2 7 and g共x兲苷兩x 1兩, find

y 4

24.

36. Center C 苷共5, 1兲, passing through 共3, 1兲

39. If f共x兲苷 x 2 4x and g共x兲苷 x 8, find

In Exercises 23 and 24, complete the graph so that it is symmetric with respect to a. the x-axis, b. the y-axis, and c. the origin. y

35. Center C 苷共2, 3兲, radius r 苷 5

38. If g共x兲苷兹64 x2, find

In Exercises 21 and 22, find the midpoint of the line segment with the given endpoints.

21. 共2, 8兲共3, 12兲

In Exercises 35 and 36, find the equation in standard form of the a circle that satisfies the given conditions.

37. If f 共x兲苷 3x 2 4x 5, find

In Exercises 19 and 20, find the distance between the points whose coordinates are given.

19. 共3, 2兲共7, 11兲

32. 兩x y兩苷 4

In Exercises 33 and 34, determine the center and radius of the circle with the given equation.

In Exercises 15 to 18, use the Pythagorean Theorem, c 2 a 2 b 2, to find the unknown. The letters a and b represent the lengths of the legs of a right triangle, and c represents the length of the hypotenuse.

23.

y2 x2 2 苷1 2 3 4

a. 共 f ⴰ g兲共5兲

b. 共g ⴰ f 兲共5兲

c. 共 f ⴰ g兲共x兲

d. 共g ⴰ f 兲共x兲

2 −4

−2

2

4 x

−4

−2

2

−2

−2

−4

−4

4 x

41. If f 共x兲苷 4x 2 3x 1, find the difference quotient f 共x h兲 f 共x兲 h

113

Review Exercises 42. If g共x兲苷 x 3 x, find the difference quotient g共x h兲 g共x兲 h

65. f 共x兲苷 3x 4

In Exercises 43 to 46, determine the domain of the function represented by the given equation.

43. f 共x兲苷 2x 2 3

44. f 共x兲苷兹6 x

45. f 共x兲苷兹25 x2

3 46. f 共x兲苷 2 x 2x 15

In Exercises 47 to 52, sketch the graph of f. Find the interval(s) on which f is increasing, constant, or decreasing.

47. f 共x兲苷兩x 3兩 2

48. f 共x兲苷 x 2 5

49. f 共x兲苷兩x 2兩兩x 2兩

50. f 共x兲苷冀x 3冁

51. f 共x兲苷

1 x3 2

In Exercises 65 to 68, find the inverse of the function. Sketch the graph of the function and its inverse on the same set of coordinate axes.

3

52. f 共x兲苷兹 x

67. h共x兲苷

66. g共x兲苷 2x 3

1 x2 2

68. k共x兲苷

1 x

69. HEIGHT OF A FALLING OBJECT The roadway of the Golden Gate Bridge is 220 feet above the water. The height h of a rock dropped from the bridge is a function of the time t it has been falling. If h is measured in feet and t is measured in seconds, then the function is given by h共t兲苷 16t 2 220. Use this function to find the time it will take the rock to hit the water. 70. BRIDGE CABLES The suspension cables of the Golden Gate Bridge approximate the shape of a parabola. If h is the height of the cables above the roadway in feet and 兩x兩 is the distance in feet from the center of the bridge, then the parabolic shape of the cables is represented by 1 2 h共x兲苷 x 25, 2100 x 2100 8820

In Exercises 53 to 58, sketch the graph of g. a. Find the domain and range of g. b. State whether g is even, odd, or neither.

53. g共x兲苷 x 2 4

54. g共x兲苷 2x 4

55. g共x兲苷兩x 2兩兩x 2兩

56. g共x兲苷兹16 x2

57. g共x兲苷 x 3 x

58. g共x兲苷 2冀x冁

1125 ft

g共x兲苷 x 3

60. f 共x兲苷 x 3 8,

g共x兲苷 x 2 2x 4

In Exercises 61 to 64, determine whether the given functions are inverses.

61. F共x兲苷 2x 5 62. h共x兲苷兹x

x5 G共x兲苷 2 k共x兲苷 x , 2

x0

x3 63. l共x兲苷 x

3 m共x兲苷 x1

x5 64. p共x兲苷 2x

2x q共x兲苷 x5

1125 ft

a. Find the height of the cables 1050 feet from the center of the bridge.

In Exercises 59 and 60, use the given functions f and g to f find f g, f g, fg, and . State the domain of each. g

59. f 共x兲苷 x 2 9,

4200 ft

b. The towers that support the cables are 2100 feet from the center of the bridge. Find the height (above the roadway) of the towers that support the cables. 71.

COMPUTER SCIENCE A test of an Internet service provider showed the following download times (in seconds) for files of various sizes (in kilobytes). Download Times Size

Time

Size

Time

10.5

0.20

110

2.01

12.9

0.24

156

2.68

15

0.27

163

2.87

20

0.36

175

3.10

60

1.09

200

3.64

75

1.42

250

4.61

a. Find a linear regression model for these data.

114

Chapter 1

b.

Functions and Graphs

From the value of r, is a linear model of the data a reasonable model? Explain.

a. Find a quadratic regression model for these data. b. On the basis of this model, will the can ever empty?

c. On the basis of the model, what is the expected download time for a file that is 100 kilobytes in size? 72.

c.

PHYSICS The rate at which water will escape from a hole in the bottom of a can depends on a number of factors, including the height of the water, the size of the hole, and the diameter of the can. The table below shows the height (in millimeters) of water in a can after t seconds.

Explain why there seems to be a contradiction between the model and reality in that we know the can will eventually run out of water.

Water Escaping a Ruptured Can Height

Time

Height

Time

0

180

60

93

10

163

70

81

20

147

80

70

30

133

90

60

40

118

100

50

50

105

110

48

» » » Quantitative Reasoning: Public Key Cryptography »»» As mentioned in the Chapter Opener, performing financial transactions over the Internet requires secure transmissions between two sites, the sender and the receiver. One method of creating secure transmissions is to use a modular function. A modular function is one that gives, in integer form, the remainder when one number is divided by another. We write a ¢ b mod m to mean that a is the remainder when b is divided by m. Here are some examples.

take note To encrypt a message means to use a secret code to change the message so that it cannot be understood by an unauthorized user. To decrypt a message means to change a coded message back to its original form.

4 ¢ 22 mod 6

because 22 6 苷 3 remainder 4.

1 ¢ 37 mod 4

because 37 4 苷 9 remainder 1.

0 ¢ 55 mod 11

because 55 11 苷 5 remainder 0.

17 ¢ 17 mod 31

because 17 31 苷 0 remainder 17.

QR1. Find the value of each expression. a. 15 mod 4 b. 37 mod 5 c. 52 mod 321 Public key cryptography uses a modular function to encrypt a message—say, a person’s name or credit card number—so that only the receiver of the message can decrypt it. The message is decrypted by using the inverse of the modular function that was used to encrypt the message. Inverse functions were discussed in Section 1.6.

Quantitative Reasoning

115

The unique feature of public key cryptography is that knowing the modular function (called the public key) used to encrypt a message does not allow one to easily determine the inverse modular function (called the private key) that is required to decrypt the message. For instance, Alice could give many people her public key to encrypt a message sent to her, but none of those people could decrypt a message written by another person and sent to Alice. QR2. You will need to do some research on public key cryptography to complete this exercise. Suppose a computer hacker wanted to decrypt a message that was encrypted using public key cryptography. What mathematical operation would the hacker need to be able to complete in order to find the inverse of the modular function that was used to encrypt the message?

take note Numbers can be assigned to letters in many ways. One common method is to use the American Standard Code for Information Interchange (ASCII). This system assigns a number to every letter, number, and punctuation mark. For instance, “A” is 65 and “a” is 97. Computing powers of a modular function is difficult without a calculator. The program POWERMOD can be used for these calculations. This program, intended for use with a T1-83/ T1-83 Plus/T1-84 Plus calculator, can be found at our website college.hmco.com/info/aufmannCAT.

To encrypt a message, first assign a number to each letter of the alphabet. For instance, let A 苷 10, B 苷 11, C 苷 12, ... , Z 苷 36. The word MATH becomes 22102917. Once a message has been converted to numbers, it is divided into blocks. We will use a block of 4 numbers so that our message is now 2210兩2917. Each block is encrypted using the public key. We will use a modulus of 3233, but in practice this number would be very large—at least 200 digits. The following are our encryption and decryption functions. Encryption function: E 苷 M 17 mod 3233, where M is the message to be coded and E is the coded message. Decryption function: M 苷 E 2753 mod 3233, where E is the message to be decoded and M is the original message. Encrypt block 1:

Encrypt block 2:

2210 mod 3233 苷 1085

291717 mod 3233 苷 904

17

Instead of 2210兩2917, the encrypted message 1085兩0904 is sent (zeros are usually inserted to keep the blocks equal in length). The person receiving the message would use the decryption function to obtain the original message. Decrypt block 1:

Decrypt block 2:

10852753 mod 3233 苷 2210

09042753 mod 3233 苷 2917

The decrypted message is 2210兩2917, or MATH. QR3. Explain how public key cryptography could be used to ensure that an email you received was sent by a particular person and not just someone using that person’s computer. If you have difficulty with this assignment, research the topic digital signatures.

116

Chapter 1

Functions and Graphs

Chapter 1 Test 1. Solve: 4x 2共2 x兲苷 5 3共2x 1兲 2. Solve: 6 3x 3 4共2 2x兲

16. Let f 共x兲苷 x 2 1 and g共x兲苷 x 2. Find 共 f g兲 and f . g

3. Solve: 2x 2 3x 苷 2

17. Find the difference quotient of the function

冉冊

f 共x兲苷 x 2 1

4. Solve: 3x 2 x 苷 2

18. Evaluate 共 f ⴰ g兲, where

5. Solve: 兩4 5x兩 6

f 共x兲苷 x 2 2x 1

6. Find the distance between the points 共2, 5兲 and 共4, 2兲.

19. Find the inverse of the function given by the equation

7. Find the midpoint and the length of the line segment with endpoints 共2, 3兲 and 共4, 1兲. 8. Determine the x- and y-intercepts of, and then graph, the equation x 苷 2y 2 4. 9. Graph the equation y 苷兩x 2兩 1.

and g共x兲苷兹x 2

f 共x兲苷

x x1

20. CALORIE CONTENT The table below shows the percent of water and the number of calories in various canned soups to which 100 grams of water are added. Percent Water in Soups % Water

Calories

% Water

Calories

93.2

28

89.6

56

11. Given f 共x兲苷兹25 x 2, evaluate f 共3兲.

92.3

26

90.5

36

91.9

39

91.9

32

12. Determine the domain of the function defined by

89.5

56

91.7

32

10. Find the center and radius of the circle that has the general form x 2 4x y 2 2y 4 苷 0.

f共x兲苷兹x 16. 2

13. Graph f 共x兲苷 2兩x 2兩 1. Identify the intervals over which the function is increasing, constant, or decreasing. 14. Use the graph of f共x兲苷兩x兩 to graph y 苷 f共x 2兲 1. 15. Which of the following define odd functions? a. f共x兲苷 x4 x 2 c. f 共x兲苷 x 1

b. f 共x兲苷 x3 x

a. Find the equation of the linear regression line for these data. b. Using the linear model from a., find the expected number of calories in a soup that is 89% water.

2

Trigonometric Functions 2.1 Angles and Arcs 2.2 Right Triangle Trigonometry 2.3 Trigonometric Functions of Any Angle 2.4 Trigonometric Functions of Real Numbers 2.5 Graphs of the Sine and Cosine Functions 2.6 Graphs of the Other Trigonometric Functions 2.7 Graphing Techniques 2.8

Harmonic Motion — An Application of the Sine and Cosine Functions

Applications of Trigonometric Functions In this chapter we introduce an important group of functions called the trigonometric functions. These functions are often used in applications involving relationships among the sides and angles of triangles. Exercise 59 on page 143 illustrates this aspect of trigonometry. In the seventeenth century, a unit circle approach was used to create trigonometric functions of real numbers or circular functions. These functions enable us to solve a wider variety of application problems. For instance, in Exercise 63 on page 178, a trigonometric function of a real number is used to model a sound wave and find the frequency of the sound wave. An application that involves finding the time it takes for the music produced by two sound tracks to repeat is given in the Quantitative Reasoning exercises on page 214. 1

Tracks

2

3

4

5

6

7

8

Guitar

Electric Bass

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

117

118

Chapter 2

Trigonometric Functions

Angles and Arcs

Section 2.1 ■ ■ ■ ■

Degree Measure Radian Measure Arcs and Arc Length Linear and Angular Speed

A point P on a line separates the line into two parts, each of which is called a halfline. The union of point P and the half-line formed by P that includes point A l l is called a ray, and it is represented as PA. The point P is the endpoint of ray PA. l

l

Figure 2.1 shows the ray PA and a second ray QR.

In geometry, an angle is defined simply as the union of two rays that have a common endpoint. In trigonometry and many advanced mathematics courses, it is beneficial to define an angle in terms of a rotation.

Definition of an Angle An angle is formed by rotating a given ray about its endpoint to some terminal position. The original ray is the initial side of the angle, and the second ray is the terminal side of the angle. The common endpoint is the vertex of the angle. R

A

There are several methods used to name an angle. One way is to employ Greek letters. For example, the angle shown in Figure 2.2 can be designated as a or as ⬔a. It also can be named ⬔O, ⬔AOB, or ⬔BOA. If you name an angle by using three points, such as ⬔AOB, it is traditional to list the vertex point between the other two points. Angles formed by a counterclockwise rotation are considered positive angles, and angles formed by a clockwise rotation are considered negative angles. See

Q P

Figure 2.1

Figure 2.3.

B α

Terminal side

O Vertex

A Initial side A

Figure 2.2

A

α O

β

O B

B

Positive angle

Negative angle

Figure 2.3

Degree Measure The measure of an angle is determined by the amount of rotation of the initial side. An angle formed by rotating the initial side counterclockwise exactly once until it coincides with itself (one complete revolution) is defined to have a measure of 360 degrees, which is abbreviated as 360°.

2.1

119

Angles and Arcs

Definition of Degree 1°

One degree is the measure of an angle formed by rotating a ray complete revolution. The symbol for degree is .

1 of a 360

The angle shown in Figure 2.4 has a measure of 1 . The angle b shown in Figure 2.5 has a measure of 30 . We will use the notation b 苷 30 to denote that the measure of angle b is 30 . The protractor shown in Figure 2.6 can be used to meas-

Figure 2.4 1 1 苷 of a revolution 360

ure an angle in degrees or to draw an angle with a given degree measure.

100 110 80 12 70 0 60 13 50 0

30 15 0

0 15 30

Figure 2.5

50 0 13

90

0 14 0 4

4 14 0 0

ββ = 30°

80 70 100 0 0 6 11 0 12

170 10

0 10 20 180 170 160

160 20

180 0

Figure 2.6 Protractor for measuring angles in degrees

Angles are often classified according to their measure. ■

180 angles are straight angles. See Figure 2.7a.

■

90 angles are right angles. See Figure 2.7b.

■

Angles that have a measure greater than 0 but less than 90 are acute angles. See Figure 2.7c.

■

Angles that have a measure greater than 90 but less than 180 are obtuse angles. See Figure 2.7d.

B B α

B

O

β

A

a. Straight angle (α = 180°)

O

B

θ

θ

O

A

b. Right angle (β = 90°)

A

c. Acute angle (0° < θ < 90°)

Figure 2.7

O

A

d. Obtuse angle (90° < θ < 180°)

120

Chapter 2

Trigonometric Functions An angle superimposed in a Cartesian coordinate system is in standard position if its vertex is at the origin and its initial side is on the positive x-axis. See

y

Figure 2.8.

Terminal side

Two positive angles are complementary angles (Figure 2.9a) if the sum of the measures of the angles is 90 . Each angle is the complement of the other angle. Two positive angles are supplementary angles (Figure 2.9b) if the sum of the measures of the angles is 180 . Each angle is the supplement of the other angle.

x Initial side

β

Figure 2.8

β

α

An angle in standard position

a. Complementary angles α + β = 90°

α

b. Supplementary angles α + β = 180°

Figure 2.9

EXAMPLE 1

u 苷 40

a.

50°

180°

Find the Measure of the Complement and the Supplement of an Angle

For each angle, find the measure (if possible) of its complement and of its supplement.

90° y

140°

»

θ = 40° x

0°

b.

u 苷 125

Solution Figure 2.10 shows ⬔u 苷 40 in standard position. The measure of its

a.

complement is 90 40 苷 50 . The measure of its supplement is 180 40 苷 140 .

Figure 2.10

Figure 2.11 shows ⬔u 苷 125 in standard position. Angle u does not have

b.

90° y

a complement because there is no positive number x such that x 125 苷 90 θ = 125°

The measure of its supplement is 180 125 苷 55 .

55° 180°

x

Figure 2.11

0°

»

Try Exercise 2, page 130

QUESTION

Are the two acute angles of any right triangle complementary angles? Explain.

Some angles have a measure greater than 360 . See Figure 2.12a and Figure 2.12b. The angle shown in Figure 2.12c has a measure less than 360 ,

ANSWER

Yes. The sum of the measures of the angles of any triangle is 180 . The right angle has a measure of 90 . Thus the sum of the measures of the two acute angles must be 180 90 苷 90 .

2.1

121

Angles and Arcs

because it is formed by a clockwise rotation of more than one revolution of the initial side. y

180° 90° x

a. 720°

c . − 990°

b. 450°

270°

Figure 2.12 Figure 2.13 90° y

β

180°

x

0°

Measures of Coterminal Angles

270°

Figure 2.14

Given ⬔u in standard position with measure x , then the measures of the angles that are coterminal with ⬔u are given by

90° y

x k 360 where k is an integer.

∠1 = 70° 180°

θ = 430° ∠2 = −290°

If the terminal side of an angle in standard position lies on a coordinate axis, then the angle is classified as a quadrantal angle. For example, the 90 angle, the 180 angle, and the 270 angle shown in Figure 2.13 are all quadrantal angles. If the terminal side of an angle in standard position does not lie on a coordinate axis, then the angle is classified according to the quadrant that contains the terminal side. For example, ⬔b in Figure 2.14 is a Quadrant III angle. Angles in standard position that have the same terminal sides are coterminal angles. Every angle has an unlimited number of coterminal angles. Figure 2.15 shows ⬔u and two of its coterminal angles, labeled ⬔1 and ⬔2.

x

0°

This theorem states that the measures of any two coterminal angles differ by an integer multiple of 360 . For instance, in Figure 2.15, u 苷 430 , ⬔1 苷 430 共1兲 360 苷 70 , and ⬔2 苷 430 共2兲 360 苷 290

270°

Figure 2.15

If we add positive multiples of 360 to 430 , we find that the angles with measures 790 , 1150 , 1510 , . . . are also coterminal with ⬔u. EXAMPLE 2

»

Classify by Quadrant and Find a Coterminal Angle

Assume the following angles are in standard position. Determine the measure of the positive angle with measure less than 360 that is coterminal with the given angle and then classify the angle by quadrant. a.

a 苷 550

b.

b 苷 225

c.

g 苷 1105

Continued 䉴

122

Chapter 2

Trigonometric Functions Solution

y

a. 190°

x

a.

Because 550 苷 190 360 , ⬔a is coterminal with an angle that has a measure of 190 . ⬔a is a Quadrant III angle. See Figure 2.16a.

b.

Because 225 苷 135 共1兲 360 , ⬔b is coterminal with an angle that has a measure of 135 . ⬔b is a Quadrant II angle. See Figure 2.16b.

c.

1105 360 苷 3

α = 550°

5 . Thus ⬔g is an angle formed by three complete 72 5 5 counterclockwise rotations, plus of a rotation. To convert 72 72 5 of a rotation to degrees, multiply times 360 . 72 5 360 苷 25 72

y

b.

135° x

β = −225°

Thus 1105 苷 25 3 360 . Hence ⬔g is coterminal with an angle that has a measure of 25 . ⬔g is a Quadrant I angle. See Figure 2.16c. y

c.

»

Try Exercise 14, page 130

25° x

γ = 1105°

Figure 2.16

There are two popular methods for representing a fractional part of a degree. One is the decimal degree method. For example, the measure 29.76 is a decimal degree. It means 29 plus 76 hundredths of 1 A second method of measurement is known as the DMS (Degree, Minute, Second) method. In the DMS method, a degree is subdivided into 60 equal parts, each of which is called a minute, denoted by . Thus 1 苷 60 . Furthermore, a minute is subdivided into 60 equal parts, each of which is called a second, denoted by . Thus 1 苷 60 and 1 苷 3600. The fractions 1 苷 1, 60

1 苷 1, 60

and

1 苷1 3600

are another way of expressing the relationships among degrees, minutes, and seconds. Each of the fractions is known as a unit fraction or a conversion factor. Because all conversion factors are equal to 1, you can multiply a numerical value by a conversion factor and not change the numerical value, even though you change the units used to express the numerical value. The following illustrates the process of multiplying by conversion factors to write 126 12 27 as a decimal degree. 126 12 27 苷 126 12 27

冉冊冉冊

苷 126 12

1 1 27 60 3600

苷 126 0.2 0.0075 苷 126.2075

2.1

123

Angles and Arcs

Integrating Technology Many graphing calculators can be used to convert a decimal degree measure to its equivalent DMS measure, and vice versa. For instance, Figure 2.17 shows that 31.57 is equivalent to 31 34 12. On a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, the degree symbol, , and the DMS function are in the ANGLE menu. ANGLE 1: ° 2: ' 3: r 4: DMS 31°34'12"

31.57° DMS

31.57

31°34'12"

Figure 2.17

Figure 2.18

To convert a DMS measure to its equivalent decimal degree measure, enter the DMS measure and press ENTER . The calculator screen in Figure 2.18 shows that 31 34 12 is equivalent to 31.57 . A TI-83/TI-83 Plus/TI-84 Plus calculator needs to be in degree mode to produce the results displayed in Figures 2.17 and 2.18. On a TI-83/TI-83 Plus/TI-84 Plus calculator, the degree symbol, , and the minute symbol, , are both in the ANGLE menu; however, the second symbol, , is entered by pressing ALPHA +. s B

Radian Measure

θ

r O

r

A

Figure 2.19

兰

Calculus Connection

Another commonly used angle measurement is the radian. To define a radian, first consider a circle of radius r and two radii OA and OB. The angle u formed by the two radii is a central angle. The portion of the circle between A and B is an arc of ២. We say that AB ២ subtends the angle u. The length of AB ២ the circle and is written AB is s (see Figure 2.19). Definition of a Radian B

One radian is the measure of the central angle subtended by an arc of length r on a circle of radius r. See Figure 2.20.

s=r

r θ

O

r

A

Figure 2.20 Central angle has a measure of 1 radian.

124

Chapter 2

Trigonometric Functions Figure 2.21 shows a protractor that can be used to measure angles in radians or to

construct angles given in radian measure.

1.8

1.6

1.4

2

1.2 1

2.2

2. 4

.8

2.6

.6

2.8

.4

3

.2

Figure 2.21 Protractor for measuring angles in radians

Definition of Radian Measure Given an arc of length s on a circle of radius r, the measure of the central s angle subtended by the arc is u 苷 radians. r

5 5

5

θ

B 5

O

5

Figure 2.22 Central angle u has a measure of 3 radians.

A

As an example, consider that an arc of length 15 centimeters on a circle with a radius of 5 centimeters subtends an angle of 3 radians, as shown in Figure 2.22. The same result can be found by dividing 15 centimeters by 5 centimeters. To find the measure in radians of any central angle u, divide the length s of the arc that subtends u by the length of the radius of the circle. Using the formula for radian measure, we find that an arc of length 12 centimeters on a circle of radius 8 centimeters subtends a central angle u whose measure is u苷

s 12 centimeters 3 radians 苷 radians 苷 radians r 8 centimeters 2

Note that the centimeter units are not part of the final result. The radian measure of a central angle formed by an arc of length 12 miles on a circle of radius 8 miles 3 would be the same, radians. If an angle has a measure of t radians, where t is a 2 real number, then the measure of the angle is often stated as t instead of t radians. For instance, if an angle u has a measure of 2 radians, we can simply write u 苷 2 instead of u 苷 2 radians. There will be no confusion concerning whether an angle measure is in degrees or radians, because the degree symbol is always used for angle measurements that are in degrees. Recall that the circumference of a circle is given by the equation C 苷 2p r. The radian measure of the central angle u subtended by the circumference is 2p r u苷苷 2p. In degree measure, the central angle u subtended by the circumferr ence is 360 . Thus we have the relationship 360 苷 2p radians. Dividing each side of the equation by 2 gives 180 苷 p radians. From this last equation, we can establish the following conversion factors.

2.1

Integrating Technology A calculator shows that 1 radian ⬇ 57.29577951

Angles and Arcs

125

Radian-Degree Conversion ■

To convert from radians to degrees, multiply by

■

To convert from degrees to radians, multiply by

and 1 ⬇ 0.017453293 radian

EXAMPLE 3

»

冉冉

冊冊

180 . p radians p radians . 180

Convert from Degrees to Radians

Convert each angle in degrees to radians. a. 60 b. 315 c. 150

Solution

冉

冊

p radians and simplify. In each case, the 180 degree units in the numerator cancel with the degree units in the denominator. Multiply each degree measure by

冉

冊

p radians 60 p p 苷 radians 苷 radians 180°/ 180 3

a.

60 苷 60°/

b.

315 苷 315°/

c.

150 苷 150°/

冉

»

冊

p radians 315 p 7p 苷 radians 苷 radians 180°/ 180 4

冉

冊冉冊

p radians 150 p 5p 苷 radians 苷 radians 180°/ 180 6

Try Exercise 32, page 130

EXAMPLE 4

»

Convert from Radians to Degrees

Convert each angle in radians to degrees. 3p 5p a. radians b. 1 radian c. radians 4 2

Solution

冉

冊

180 and simplify. In each case, the p radians radian units in the numerator cancel with the radian units in the denominator. Multiply each radian measure by

冉

冊冉

a.

3p 3p radians radians 苷 4 4

b.

1 radian 苷共1 radian兲

c.

»

冉

冉

冊

180° 3 180° 苷苷 135° p radians 4

冊冊冉

180° 180° 苷 ⬇ 57.3° p radians p

5p 5p radians radians 苷 2 2

Try Exercise 44, page 130

冊

180° 5 180° 苷苷 450° p radians 2

126

Chapter 2

Table 2.1 Degrees

Radians

Trigonometric Functions Table 2.1 lists the degree and radian measures of selected angles. Figure 2.23 illustrates each angle listed in the table as measured from the positive x-axis. π 90°, 2

0

0

30

兾6

45

兾4

60

兾3

90

兾2

120

2兾3

135

3兾4

150

5兾6

180

210

7兾6

225

5兾4

240

4兾3

Figure 2.23

270

3兾2

Degree and radian measures of selected angles

300

5兾3

315

7兾4

330

11兾6

360

2

100° 1.745329252

Figure 2.24 2.2r 126.0507149

2π 120°, 3 135°, 3π 4 150°, 5π 6

60°,

π 3

45°,

π 4

30°,

180°, π

0°, 0

210°, 7π 6 225°, 5π 4 240°, 4π 3

π 6

330°,

270°, 3π 2

11π 6

7π 315°, 4 5π 300°, 3

Integrating Technology A graphing calculator can convert degree measure to radian measure, and vice versa. For example, the calculator display in Figure 2.24 shows that 100 is approximately 1.74533 radians. The calculator must be in radian mode to convert from degrees to radians. The display in Figure 2.25 shows that 2.2 radians is approximately 126.051 . The calculator must be in degree mode to convert from radians to degrees. On a TI-83/TI-83 Plus/TI-84 Plus calculator, the symbol for radian measure is r, and it is in the ANGLE menu.

Arcs and Arc Length Consider a circle of radius r. By solving the formula u 苷

s for s, we have an equar

tion for arc length. Arc Length Formula

Figure 2.25

Let r be the length of the radius of a circle and u the nonnegative radian measure of a central angle of the circle. Then the length of the arc s that subtends the central angle is s 苷 ru. See Figure 2.26.

s = arc length B θ

O

r

Figure 2.26 s 苷 ru

A

2.1

EXAMPLE 5

»

Angles and Arcs

127

Find the Length of an Arc

Find the length of an arc that subtends a central angle of 120 in a circle of radius 10 centimeters.

Solution

take note The formula s 苷 r u is valid only

The formula s 苷 ru requires that u be expressed in radians. We first convert 120 to radian measure and then use the formula s 苷 ru.

冉

when u is expressed in radians.

u 苷 120 苷 120

冊冉冊

p radians 2p 2p 苷 radians 苷 180 3 3

s 苷 ru 苷共10 centimeters兲

2p 20p 苷 centimeters 3 3

»

Try Exercise 68, page 130

EXAMPLE 6

»

Solve an Application

A pulley with a radius of 10 inches uses a belt to drive a pulley with a radius of 6 inches. Find the angle through which the smaller pulley turns as the 10-inch pulley makes one revolution. State your answer in radians and also in degrees.

Solution S1

θ1

10 in.

Use the formula s 苷 ru. As the 10-inch pulley turns through an angle u1, a point on the rim of that pulley moves s1 inches, where s1 苷 10u1. See Figure 2.27. At the same time, the 6-inch pulley turns through an angle of u2 and a point on the rim of that pulley moves s2 inches, where s2 苷 6u 2. Assuming that the belt does not slip on the pulleys, we have s1 苷 s2. Thus 10u1 苷 6u2 10共2p兲苷 6u2

S2

θ2

• Solve for 2 when 1 2 radians.

10 p 苷 u2 3 6 in.

The 6-inch pulley turns through an angle of Figure 2.27

10 p radians, or 600 . 3

»

Try Exercise 72, page 131

Linear and Angular Speed A ride at an amusement park has an inner ring of swings and an outer ring of swings. During each complete revolution, the swings in the outer ring travel a greater distance than the swings in the inner ring. We can say that the swings in the outer ring have a greater linear speed than the swings in the inner ring. Interestingly enough, all of the swings complete the same number of revolutions

128

Chapter 2

Trigonometric Functions during any given ride. We say that all of the swings have the same angular speed. In the following definitions, v denotes linear speed and v (omega) denotes angular speed. Definition of Linear and Angular Speed of a Point Moving on a Circular Path A point moves on a circular path with radius r at a constant rate of u radians per unit of time t. Its linear speed is v苷

s t

where s is the distance the point travels, given by s 苷 ru. The point’s angular speed is v苷

u t

Some common units of angular speed are revolutions per second, revolutions per minute, radians per second, and radians per minute.

EXAMPLE 7

»

Convert an Angular Speed

A hard disk in a computer rotates at 3600 revolutions per minute. Find the angular speed of the disk in radians per second.

Solution As a point on the disk rotates 1 revolution, the angle through which the 2p radians point moves is 2p radians. Thus will be the conversion factor 1 revolution we will use to convert from revolutions to radians. To convert from minutes 1 minute to seconds, use the conversion factor . 60 seconds

冉

冊冉

冊

3600 revolutions 2p radians 1 minute 1 minute 1 revolution 60 seconds 苷 120p radians兾second • Exact answer • Approximate answer ⬇ 377 radians兾second

3600 revolutions兾minute 苷

»

Try Exercise 74, page 131

We can establish an important relationship between linear speed and angular speed. We start with the linear speed formula and then substitute ru for s, as shown below. v苷

s ru u 苷苷 r 苷 rv t t t

2.1

Angles and Arcs

129

Thus the linear speed of a point moving on a circular path is the product of the radius of the circle and the angular speed of the point.

The Linear Speed–Angular Speed Relationship The linear speed v and the angular speed v, in radians per unit of time, of a point moving on a circular path with radius r are related by v 苷 rv

The equation v 苷 rv gives the linear speed of a point moving on a circular path in terms of distance r from the center of the circle and the angular speed v, provided v is in radians per unit of time.

EXAMPLE 8

»

Find a Linear Speed

A wind machine is used to generate electricity. The wind machine has propeller blades that are 12 feet in length (see Figure 2.28). If the propeller is rotating at 3 revolutions per second, what is the linear speed in feet per second of the tips of the blades?

Solution 12 ft

Convert the angular speed v 苷 3 revolutions per second into radians per second, and then use the formula v 苷 rv. v苷

冉

冊冉

3 revolutions 3 revolutions 苷 1 second 1 second

Thus

冉

冊

2p radians 6p radians 苷 1 revolution 1 second

冊

6p radians 1 second 苷 72p feet per second ⬇ 226 feet per second

v 苷 rv 苷共12 feet兲

Figure 2.28

»

Try Exercise 80, page 131

Topics for Discussion 1. The measure of a radian differs depending on the length of the radius of the circle used. Do you agree? Explain. 2. What are the necessary conditions for an angle to be in standard position? 3. Is the supplement of an obtuse angle an acute angle? 4. Do all acute angles have a positive measure?

130

Chapter 2

Trigonometric Functions

Exercise Set 2.1 In Exercises 1 to 12, find the measure (if possible) of the complement and the supplement of each angle.

1. 15

2. 87

3. 70 15

4. 22 43

5. 56 33 15

6. 19 42 05

7. 1

8. 0.5

9.

p 4

2p 5

12.

p 6

10.

p 3

11.

In Exercises 13 to 18, determine the measure of the positive angle with measure less than 360° that is coterminal with the given angle and then classify the angle by quadrant. Assume the angles are in standard position.

13. a 苷 610

14. a 苷 765

15. a 苷 975

16. a 苷 872

17. a 苷 2456

18. a 苷 3789

In Exercises 19 to 24, use a calculator to convert each decimal degree measure to its equivalent DMS measure.

46.

2p 3

47.

p 6

48.

p 9

11p 18

51.

11p 3

49.

3p 8

50.

52.

6p 5

5p 53. 12

4p 54. 5

In Exercises 55 to 60, convert radians to degrees or degrees to radians. Round answers to the nearest hundredth.

55. 1.5

56. 2.3

57. 133

58. 427

59. 8.25

60. 90

In Exercises 61 to 64, find the measure in radians and degrees of the central angle of a circle subtended by the given arc. Round approximate answers to the nearest hundredth.

61. r 苷 2 inches, s 苷 8 inches 62. r 苷 7 feet, s 苷 4 feet

19. 24.56

20. 110.24

21. 64.158

63. r 苷 5.2 centimeters, s 苷 12.4 centimeters

22. 18.96

23. 3.402

24. 224.282

64. r 苷 35.8 meters, s 苷 84.3 meters

In Exercises 25 to 30, use a calculator to convert each DMS measure to its equivalent decimal degree measure.

25. 25 25 12

26. 63 29 42

27. 183 33 36

28. 141 6 9

29. 211 46 48

30. 19 12 18

In Exercises 31 to 42, convert the degree measure to exact radian measure.

In Exercises 65 to 68, find the length of an arc that subtends a central angle with the given measure in a circle with the given radius. Round answers to the nearest hundredth.

65. r 苷 8 inches, u 苷

66. r 苷 3 feet, u 苷

p 4

7p 2

31. 30

32. 45

33. 90

67. r 苷 25 centimeters, u 苷 42

34. 15

35. 165

36. 315

68. r 苷 5 meters, u 苷 144

37. 420

38. 630

39. 585

40. 135

41. 9

42. 110

In Exercises 43 to 54, convert the radian measure to exact degree measure.

43.

7p 3

44.

p 4

45.

p 5

69. Find the number of radians in 1

70. Find the number of radians in

1 revolutions. 2

3 revolution. 8

71. ANGULAR ROTATION OF TWO PULLEYS A pulley with a radius of 14 inches uses a belt to drive a pulley with a radius of

2.1 28 inches. The 14-inch pulley turns through an angle of 150°. Find the angle through which the 28-inch pulley turns. 72. ANGULAR ROTATION OF TWO PULLEYS A pulley with a diameter of 1.2 meters uses a belt to drive a pulley with a diameter of 0.8 meter. The 1.2-meter pulley turns through an angle of 240°. Find the angle through which the 0.8-meter pulley turns. 73. ANGULAR SPEED Find the angular speed, in radians per second, of the second hand on a clock. 74. ANGULAR SPEED Find the angular speed, in radians per second, of a point on the equator of the earth. 75. ANGULAR SPEED A wheel is rotating at 50 revolutions per minute. Find the angular speed in radians per second. 76. ANGULAR SPEED A wheel is rotating at 200 revolutions per minute. Find the angular speed in radians per second. 77. ANGULAR SPEED The turntable of a record player turns 1 at 33 revolutions per minute. Find the angular speed in 3 radians per second. 78. ANGULAR SPEED A car with a wheel of radius 14 inches is moving with a speed of 55 mph. Find the angular speed of the wheel in radians per second. 79. LINEAR SPEED OF A CAR Each tire on a car has a radius of 15 inches. The tires are rotating at 450 revolutions per minute. Find the speed of the automobile to the nearest mile per hour. 80. LINEAR SPEED OF A TRUCK Each tire on a truck has a radius of 18 inches. The tires are rotating at 500 revolutions per minute. Find the speed of the truck to the nearest mile per hour. 81. BICYCLE GEARS The chain wheel of Emma’s bicycle has a radius of 3.5 inches. The rear gear has a radius of 1.75 inches, and the back tire has a radius of 12 inches. If Emma pedals for 150 revolutions of the chain wheel, how far will she travel? Round to the nearest foot.

82. ROTATION VERSUS LIFT DISTANCE A winch with a 6-inch radius is used to lift a container. The winch is designed so that as it is rotated, the cable stays in contact with the surface of the winch. That is, the cable does not wrap on top of itself.

Radius 6 in.

FRA

a. Find the distance the container is lifted as the winch is rotated through an angle 5p of radians. 6

GILE

L ND HA ITH W RE CA

E

b. Determine the angle, in radians, through which the winch must be rotated to lift the container a distance of 2 feet. 83. AMUSEMENT PARK RIDE A ride at an amusement park consists of two circular rings of swings. At full speed the swings in the inner ring travel on a circular path with a radius of 32 feet and the swings in the outer ring travel on a circular path with a radius of 38 feet. Each swing makes one complete revolution every 3.75 seconds. How much greater, in miles per hour, is the linear speed of the swings in the outer ring than the linear speed of the swings in the inner ring? Round to the nearest tenth of a mile per hour. 84. HORSE RACING The semicircular turns of a horse race track each have a radius of 200 feet. During the first turn of a race, the lead horse is running near the inside rail on a path with a 202.0-foot radius, at a constant rate of 24.4 feet per second. A second horse is rounding the same turn on a path with a 206.5-foot radius. At what constant rate does the second horse need to run to keep pace with the lead horse during this turn? Round to the nearest tenth of a foot per second. 85. ASTRONOMY At a time when Earth was 93,000,000 miles from the sun, you observed through a tinted glass that the diameter of the sun occupied an arc of 31 . Determine, to the nearest ten thousand miles, the diameter of the sun.

93,000,000 miles

A

31'

Radius 1.75 in.

131

Angles and Arcs

Sun

Earth B

Radius 12 in.

Radius 3.5 in.

(Hint: Because the radius of arc AB is large and its central angle is small, the length of the diameter of the sun is approximately the length of the arc AB.)

132

Chapter 2

Trigonometric Functions Eratosthenes reasoned that because the sun is far away, the rays of sunlight that reach Earth must be nearly parallel. From this assumption he concluded that the measure of ⬔AOS in the accompanying figure must be 7.5 . Use this information to estimate the radius (to the nearest 10 miles) of Earth.

86. ANGLE OF ROTATION AND DISTANCE The minute hand on the clock atop city hall measures 6 feet 3 inches from its tip to its axle. a. Through what angle (in radians) does the minute hand pass between 9:12 A.M. and 9:48 A.M.? b. What distance, to the nearest tenth of a foot, does the tip of the minute hand travel during this period? 87. VELOCITY OF THE HUBBLE SPACE TELESCOPE On April 25, 1990, the Hubble Space Telescope (HST) was deployed into a circular orbit 625 kilometers above the surface of the earth. The HST completes an Earth orbit every 1.61 hours.

89.

VELOCITY COMPARISONS Assume that the bicycle in the figure is moving forward at a constant rate. Point A is on the edge of the 30-inch rear tire, and point B is on the edge of the 20-inch front tire.

A

B

a. Which point (A or B) has the greater angular velocity? b. Which point (A or B) has the greater linear velocity?

a. Find the angular velocity, with respect to the center of Earth, of the HST. Round your answer to the nearest 0.1 radian per hour. b. Find the linear velocity of the HST. (Hint: The radius of Earth is about 6370 kilometers.) Round your answer to the nearest 100 kilometers per hour. 88. ESTIMATING THE RADIUS OF EARTH Eratosthenes, the fifth librarian of Alexandria (230 B.C.), was able to estimate the radius of Earth from the following data: The distance between the Egyptian cities of Alexandria and Syrene was 5000 stadia (520 miles). Syrene was located directly south of Alexandria. One summer, at noon, the sun was directly overhead at Syrene, whereas at the same time in Alexandria, the sun was at a 7.5 angle from the zenith.

7.5° Alexandria

Syrene A

S 7.5°

O

90. Given that s, r, u, t, v, and v are as defined in Section 2.1, determine which of the following formulas are valid. s 苷 ru

r苷

s u

v苷

ru t

v 苷 rv

v苷

s t

v苷

u t

91. NAUTICAL MILES AND STATUTE MILES A nautical mile is the length of an arc, on Earth’s equator, that subtends a 1 central angle. The equatorial radius of Earth is about 3960 statute miles. a. Convert 1 nautical mile to statute miles. Round to the nearest hundredth of a statute mile. b. Determine what percent (to the nearest 1 percent) of Earth’s circumference is covered by a trip from Los Angeles, California to Honolulu, Hawaii (a distance of 2217 nautical miles). 92. PHOTOGRAPHY The field of view for a camera with a 200-millimeter lens is 12 . A photographer takes a photograph of a large building that is 485 feet in front of the camera. What is the approximate width, to the nearest foot, of the building that will appear in the photograph? (Hint: If the radius of an arc AB is large and its central angle is small, then the length of the line segment AB is approximately the length of the arc AB.)

2.1

133

Angles and Arcs

Connecting Concepts A sector of a circle is the region bounded by radii OA and OB and the intercepted arc AB. See the following figure. The B area of the sector is given by 1 2 r 2

EARTH radius 3960 miles

r θ

O

N

A

eri d i

an

where r is the radius of the circle and is the measure of the central angle in radians.

r

New York City 40° 45’ N

In Exercises 93 to 96, find the area, to the nearest square unit, of the sector of a circle with the given radius and central angle.

Miami 25° 47’ N

r

Greenwich M

A

tude of 0. The north pole has a latitude of 90. The radius of Earth is approximately 3960 miles.

Equator

p 93. r 苷 5 inches, u 苷 radians 3 94. r 苷 2.8 feet, u 苷

Longitude east

5p radians 2

Latitude north

97. GEOGRAPHY The city of Miami has a latitude of 25 47 N. How far north, to the nearest 10 miles, is it from the equator? Use 3960 miles as the radius of Earth.

95. r 苷 120 centimeters, u 苷 0.65 radian 96. r 苷 30 feet, u 苷 62 Latitude describes the position of a point on Earth’s surface in relation to the equator. A point on the equator has a lati-

98. GEOGRAPHY New York City has a latitude of 40 45 N. How far north, to the nearest 10 miles, is it from the equator? Use 3960 miles as the radius of Earth.

Projects 1. CURRENCY CONVERSION You are traveling in a foreign country. You discover that the currency used consists of lollars, mollars, nollars, and tollars.

d. If you wish to convert x degrees to radians, you need to multiply x by which of the following conversion factors?

冉冊冉冊 p 180

5 lollars 苷 1 mollar 3 mollars 苷 5 nollars 4 nollars 苷 7 tollars The fare for a taxi is 14 tollars. a. How much is the fare in mollars? (Hint: Make use of conversion factors to convert from tollars to mollars.) b. If you have only mollars and lollars, how many of each could you use to pay the fare? c.

Explain how you knew which of the following conversion factors should be used in the conversion in a.

冉

冊冉

4 nollars 7 tollars

冊

7 tollars 4 nollars

180 p

2. SPACE SHUTTLE The rotational period of Earth is 23.933 hours. A space shuttle revolves around Earth’s equator every 2.231 hours. Both are rotating in the same direction. At the present time, the space shuttle is directly above the Galápagos Islands. How long will it take for the space shuttle to circle Earth and return to a position directly above the Galápagos Islands?

Galápagos Islands

S P Space shuttle

Equator

134

Chapter 2

Section 2.2 ■

■

■

The Six Trigonometric Functions Trigonometric Functions of Special Angles Applications Involving Right Triangles

Trigonometric Functions

Right Triangle Trigonometry P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A9.

PS1. Rationalize the denominator of

PS2. Rationalize the denominator of

冉冊冉冊冉冊

PS3. Simplify: a

a 2

a 2

PS4. Simplify:

1 兹3 2 兹2

.

.

兹3 a 2

PS5. Solve

x 兹2 苷 for x. Round your answer to the nearest hundredth. [1.1] 2 5

PS6. Solve

x 兹3 苷 for x. Round your answer to the nearest hundredth. [1.1] 3 18

The Six Trigonometric Functions The study of trigonometry, which means “triangle measurement,” began more than 2000 years ago, partially as a means of solving surveying problems. Early trigonometry used the length of a line segment between two points of a circle as the value of a trigonometric function. In the sixteenth century, right triangles were used to define a trigonometric function. We will use a modification of this approach. When working with right triangles, it is convenient to refer to the side opposite an angle or the side adjacent to (next to) an angle. Figure 2.29 shows the sides opposite and adjacent to the angle u. Figure 2.30 shows the sides opposite and adjacent to the angle b. In both cases, the hypotenuse remains the same.

Hypotenuse

Opposite side

Hypotenuse

β

Adjacent side

θ

Adjacent side

Adjacent and opposite sides of ∠q

Figure 2.29

Opposite side

Adjacent and opposite sides of ∠β

Figure 2.30

Six ratios can be formed by using two lengths of the three sides of a right triangle. Each ratio defines a value of a trigonometric function of a given acute angle . The functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

2.2

135

Right Triangle Trigonometry

Definitions of Trigonometric Functions of an Acute Angle Let be an acute angle of a right triangle. See Figure 2.29. The values of the six trigonometric functions of are length of opposite side length of hypotenuse length of opposite side tan u 苷 length of adjacent side length of hypotenuse sec u 苷 length of adjacent side

sin u 苷

length of adjacent side length of hypotenuse length of adjacent side cot u 苷 length of opposite side length of hypotenuse csc u 苷 length of opposite side

cos u 苷

We will write opp, adj, and hyp as abbreviations for the length of the opposite side, adjacent side, and hypotenuse, respectively.

EXAMPLE 1

»

Evaluate Trigonometric Functions

Find the values of the six trigonometric functions of for the triangle given in Figure 2.31.

Solution

hyp

3

Use the Pythagorean Theorem to find the length of the hypotenuse. hyp 苷兹32 42 苷兹25 苷 5

θ

4

From the definitions of the trigonometric functions,

Figure 2.31

opp 苷 hyp adj cot 苷苷 opp

sin 苷

3 5 4 3

adj 苷 hyp hyp sec 苷苷 adj

cos 苷

4 5 5 4

opp 苷 adj hyp csc 苷苷 opp

tan 苷

3 4 5 3

»

Try Exercise 6, page 142

Given the value of one trigonometric function of the acute angle , it is possible to find the value of any of the remaining trigonometric functions of .

EXAMPLE 2

»

Find the Value of a Trigonometric Function

Given that is an acute angle and cos 苷

5 , find tan . 8 Continued 䉴

136

Chapter 2

Trigonometric Functions Solution cos 苷

5 adj 苷 8 hyp

Sketch a right triangle with one leg of length 5 units and a hypotenuse of length 8 units. Label as the acute angle that has the leg of length 5 units as its adjacent side (see Figure 2.32). Use the Pythagorean Theorem to find the length of the opposite side.

8

opp

共opp兲2 52 苷 82 共opp兲2 25 苷 64 共opp兲2 苷 39 opp 苷兹39

θ

5

Figure 2.32

Therefore, tan 苷

opp 兹39 苷 . adj 5

»

Try Exercise 18, page 142

Trigonometric Functions of Special Angles In Example 1, the lengths of the legs of the triangle were given, and you were asked to find the values of the six trigonometric functions of the angle . Often we will want to find the value of a trigonometric function when we are given the measure of an angle rather than the measure of the sides of a triangle. For most angles, advanced mathematical methods are required to evaluate a trigonometric function. For some special angles, however, the value of a trigonometric function can be found by geometric methods. These special acute angles are 30 , 45 , and 60 . First, we will find the values of the six trigonometric functions of 45 . (This discussion is based on angles measured in degrees. Radian measure could have been used without changing the results.) Figure 2.33 shows a right triangle with angles 45 , 45 , and 90 . Because ⬔A 苷 ⬔B, the lengths of the sides opposite these angles are equal. Let the length of each equal side be denoted by a. From the Pythagorean Theorem,

B 45° r= 2a

a

45° a

A

C

Figure 2.33

r 2 苷 a2 a2 苷 2a2 r 苷兹2a2 苷兹2 a The values of the six trigonometric functions of 45 are A

sin 45 苷

30°

C

60°

O

a 2

Figure 2.34

B

cos 45 苷

a 1 兹2 苷苷 2 兹2 a 兹2

tan 45 苷

a 苷1 a

cot 45 苷

a 苷1 a

sec 45 苷

兹2 a 苷兹2 a

csc 45 苷

兹2 a 苷兹2 a

a 3a h= 2

1 a 兹2 苷苷 2 兹2 a 兹2

The values of the trigonometric functions of the special angles 30 and 60 can be found by drawing an equilateral triangle and bisecting one of the angles, as Figure 2.34 shows. The angle bisector also bisects one of the sides. Thus the length of the side opposite the 30 angle is one-half the length of the hypotenuse of triangle OAB.

2.2

137

Right Triangle Trigonometry

Let a denote the length of the hypotenuse. Then the length of the side oppoa site the 30 angle is . The length of the side adjacent to the 30 angle, h, is found 2 by using the Pythagorean Theorem. a2 苷

冉冊

a2 苷

a2 h2 4

a 2

2

h2

3a2 苷 h2 4 h苷

• Subtract

兹3a 2

a2 from each side. 4

• Solve for h.

The values of the six trigonometric functions of 30 are sin 30 苷

a兾2 1 苷 a 2

cos 30 苷

兹3 a兾2 兹3 苷 a 2

tan 30 苷

a兾2 1 兹3 苷苷 3 兹3 a兾2 兹3

cot 30 苷

兹3 a兾2 苷兹3 a兾2

sec 30 苷

a 2 2 兹3 苷苷 3 a兾2 兹3 兹3

csc 30 苷

a 苷2 a兾2

The values of the trigonometric functions of 60 can be found by again using 兹3a Figure 2.34. The length of the side opposite the 60 angle is , and the length 2 a of the side adjacent to the 60 angle is . The values of the trigonometric func2 tions of 60 are a兾2 1 兹3 a兾2 兹3 sin 60 苷苷 cos 60 苷苷 a 2 a 2 兹3 a兾2 苷兹3 a兾2 a sec 60 苷苷2 a兾2

tan 60 苷

a兾2 1 兹3 苷苷 3 兹3 a兾2 兹3 a 2 2 兹3 csc 60 苷苷苷 3 兹3 a兾2 兹3

cot 60 苷

Table 2.2 summarizes the values of the trigonometric functions of the special

angles 30 共兾6兲, 45 共兾4兲, and 60 共兾3兲. Table 2.2

take note

Memorizing the values given in Table 2.2 will prove to be extremely

sin

cos

tan

csc

sec

cot

30 ;

6

1 2

兹3 2

兹3 3

2

2兹3 3

兹3

45 ;

4

兹2 2

兹2 2

1

兹2

兹2

1

60 ;

3

兹3 2

1 2

兹3

2兹3 3

2

兹3 3

useful in the remaining trigonometry sections.

Trigonometric Functions of Special Angles

138

Chapter 2

Trigonometric Functions What is the measure, in degrees, of the acute angle for which sin 苷 cos , tan 苷 cot , and sec 苷 csc ?

QUESTION

EXAMPLE 3

»

Evaluate a Trigonometric Expression

Find the exact value of sin2 45 cos2 60 . Note: sin2 苷共sin 兲共sin 兲苷共sin 兲2 and cos2 苷共cos 兲共cos 兲苷共cos 兲2.

take note The patterns in the following chart can be used to memorize the sine and cosine of 30°, 45°, and 60°. sin 30 苷

兹1 2

cos 30 苷

兹3 2

sin 45 苷

兹2 2

cos 45 苷

兹2 2

sin 60 苷

兹3 2

cos 60 苷

兹1 2

Solution Substitute the values of sin 45 and cos 60 and simplify. sin2 45 cos2 60 苷

冉冊冉冊兹2 2

2

1 2

2

苷

2 1 3 苷 4 4 4

»

Try Exercise 34, page 142

From the definition of the sine and cosecant functions, opp hyp 共sin u兲共csc u兲苷苷1 or 共sin 兲共csc 兲苷 1 hyp opp By rewriting the last equation, we find 1 1 sin 苷 and csc 苷 , provided sin 苷 0 csc sin The sine and cosecant functions are called reciprocal functions. The cosine and secant are also reciprocal functions, as are the tangent and cotangent functions. Table 2.3 shows each trigonometric function and its reciprocal. These relationships hold for all values of for which both of the functions are defined. Table 2.3 Trigonometric Functions and Their Reciprocals sin 苷

1 csc

cos 苷

1 sec

tan 苷

1 cot

csc 苷

1 sin

sec 苷

1 cos

cot 苷

1 tan

Integrating Technology The values of the trigonometric functions of the special angles 30 , 45 , and 60 shown in Table 2.2 are exact values. If an angle is not one of these special angles, then a graphing calculator often is used to approximate the value of a trigonometric function. For instance, to find sin 52.4 on a TI-83/TI-83 Plus/TI-84 Plus calculator, first check that the calculator is in degree mode. Then use the sine function key SIN to key in sin(52.4) and press ENTER . See Figure 2.35. ANSWER

45°

2.2

Right Triangle Trigonometry

139

To find sec 1.25, first check that the calculator is in radian mode. A TI-83/ TI-83 Plus/TI-84 Plus calculator does not have a secant function key, but because the secant function is the reciprocal of the cosine function, we can evaluate sec 1.25 by evaluating 1/(cos 1.25). See Figure 2.36. Select Degree in the Mode menu.

Select Radian in the Mode menu.

Normal Sci Eng Float 0123456789 Radian Degree Func sin(52.4) Cong .7922896434 Sequ Real Full

Normal Sci Eng Float 0123456789 Radian Degree Func 1/(cos(1.25)) Cong 3.171357694 Sequ Real Full

Figure 2.35

Figure 2.36

When you evaluate a trigonometric function with a calculator, be sure the calculator is in the correct mode. Many errors are made because the correct mode has not been selected. Line of sight

Angle of elevation

Horizontal line

Angle of depression Line of sight

Applications Involving Right Triangles Some applications concern an observer looking at an object. In these applications, angles of elevation or angles of depression are formed by a line of sight and a horizontal line. If the object being observed is above the observer, the acute angle formed by the line of sight and the horizontal line is an angle of elevation. If the object being observed is below the observer, the acute angle formed by the line of sight and the horizontal line is an angle of depression. See Figure 2.37.

Figure 2.37

EXAMPLE 4

»

Solve an Angle-of-Elevation Application

From a point 115 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 64.3 . Find the height of the tree to the nearest foot.

Solution From Figure 2.38, the length of the adjacent side of the angle is known (115 feet). Because we need to determine the height of the tree (length of the opposite side), we use the tangent function. Let h represent the length of the opposite side. h

opp h 苷 adj 115 h 苷 115 tan 64.3 ⬇ 238.952

tan 64.3 苷 64.3° 115 ft

Figure 2.38

The height of the tree is approximately 239 feet.

»

Try Exercise 56, page 143

• Use a calculator to evaluate tan 64.3°.

140

Chapter 2

Trigonometric Functions

take note The significant digits of an

Because the cotangent function involves the sides adjacent to and opposite an angle, we could have solved Example 4 by using the cotangent function. The solution would have been

approximate number are

adj 115 苷 opp h 115 h苷 ⬇ 238.952 feet cot 64.3

cot 64.3 苷

every nonzero digit. the digit 0, provided it is between two nonzero digits or it is to the right of a nonzero digit in a number which includes a decimal point. For example, the approximate number:

The accuracy of a calculator is sometimes beyond the limits of measurement. In Example 4 the distance from the base of the tree was given as 115 feet (three significant digits), whereas the height of the tree was shown to be 238.952 feet (six significant digits). When using approximate numbers, we will use the conventions given below for calculating with trigonometric functions.

502 has 3 significant digits. 3700 has 2 significant digits. 47.0 has 3 significant digits. 0.0023 has 2 significant digits.

A Rounding Convention: Significant Digits for Trigonometric Calculations

0.00840 has 3 significant digits.

Angle Measure to the Nearest

Significant Digits of the Lengths

Degree

Two

Tenth of a degree

Three

Hundredth of a degree

Four

EXAMPLE 5

»

Solve an Angle-of-Depression Application

DME (Distance Measuring Equipment) is standard avionic equipment on a commercial airplane. This equipment measures the distance from a plane to a radar station. If the distance from a plane to a radar station is 160 miles and the angle of depression is 33 , find the number of ground miles from a point directly below the plane to the radar station.

Solution From Figure 2.39, the length of the hypotenuse is known (160 miles). The length of the side opposite the angle of 57 is unknown. The sine function involves the hypotenuse and the opposite side, x, of the 57 angle.

33° 57°

x 160 x 苷 160 sin 57 ⬇ 134.1873

160 mi

sin 57 苷

x

Rounded to two significant digits, the plane is 130 ground miles from the radar station. Figure 2.39

»

Try Exercise 58, page 143

2.2

EXAMPLE 6

»

Right Triangle Trigonometry

141

Solve an Angle-of-Elevation Application

An observer notes that the angle of elevation from point A to the top of a space shuttle is 27.2 . From a point 17.5 meters further from the space shuttle, the angle of elevation is 23.9 . Find the height of the space shuttle.

Solution y

23.9° 27.2°

From Figure 2.40, let x denote the distance from point A to the base of the space shuttle, and let y denote the height of the space shuttle. Then

x

(1) tan 27.2 苷

A

y x

17.5 m

Solving Equation (1) for x, x 苷 Figure 2.40

and

(2) tan 23.9 苷

y x 17.5

y 苷 y cot 27.2 , and substituting tan 27.2

into Equation (2), we have y y cot 27.2 17.5 y 苷共tan 23.9 兲共y cot 27.2 17.5兲

tan 23.9 苷

take note The intermediate calculations in

• Solve for y.

y y tan 23.9 cot 27.2 苷共tan 23.9 兲共17.5兲

Example 6 were not rounded off. This ensures better accuracy for

y(1 tan 23.9 cot 27.2 ) 苷共tan 23.9 兲共17.5兲共tan 23.9 兲共17.5兲 y苷 ⬇ 56.2993 1 tan 23.9 cot 27.2

the final result. Using the rounding convention stated on page 140, we round off only the last result.

To three significant digits, the height of the space shuttle is 56.3 meters.

»

Try Exercise 66, page 145

Topics for Discussion 1. If is an acute angle of a right triangle for which cos 苷 the case that sin 苷

3 , then it must be 8

5 . Do you agree? Explain. 8

2. A tutor claims that tan 30 苷 cot 60 . Do you agree? 3. Does sin 2 苷 2 sin ? Explain. 4. How many significant digits are in each of the following measurements? a.

0.0042 inches

b. 5.03 inches

c. 62.00 inches

142

Chapter 2

Trigonometric Functions

Exercise Set 2.2 In Exercises 1 to 12, find the values of the six trigonometric functions of for the right triangle with the given sides.

1.

2.

In Exercises 16 to 18, let be an acute angle of a right trian4 gle for which tan . Find 3

17. cot

16. sin

18. sec

7

12

In Exercises 19 to 21, let be an acute angle of a right 13 triangle for which sec . Find 12

θ

θ

3

5

20. cot

19. cos

3.

4. 7

9

4

3

θ

θ

5.

In Exercises 22 to 24, let be an acute angle of a right tri2 angle for which cos . Find 3

23. sec

22. sin

6. 8

5

5 θ θ

2

21. csc

24. tan

In Exercises 25 to 38, find the exact value of each expression.

25. sin 45 cos 45

26. csc 45 sec 45

27. sin 30 cos 60 tan 45

28. csc 60 sec 30 cot 45

29. sin 30 cos 60 tan 45 7.

8. 3

θ

5

2

9.

θ

6

10

30. sec 30 cos 30 tan 60 cot 60

θ

31. sin

cos 3 6

32. csc

sec 6 3

33. sin

tan 4 6

34. sin

cos tan 3 4 4

35. sec

cos tan 3 3 6

36. cos

tan 2 tan 4 6 3

10. θ

3 0.8

11.

1

12. θ

2

5

θ

1

38. 3 tan

37. 2 csc

sec cos 4 3 6

sec sin 4 6 3

6

In Exercises 13 to 15, let be an acute angle of a right 3 triangle for which sin . Find 5

13. tan

14. sec

15. cos

In Exercises 39 to 50, use a calculator to find the value of the trigonometric function to four decimal places.

39. tan 32

40. sec 88

41. cos 63 20

42. cot 55 50

43. cos 34.7

44. tan 81.3

2.2 45. sec 5.9

48. sec

46. sin

3 8

5

47. tan

49. csc 1.2

7

50. sin 0.45

51. VERTICAL HEIGHT FROM SLANT HEIGHT A 12-foot ladder is resting against a wall and makes an angle of 52 with the ground. Find the height to which the ladder will reach on the wall.

Right Triangle Trigonometry

143

55. PLACEMENT OF A LIGHT For best illumination of a piece of art, a lighting specialist for an art gallery recommends that a ceiling-mounted light be 6 feet from the piece of art and that the angle of depression of the light be 38 . How far from a wall should the light be placed so that the recommendations of the specialist are met? Notice that the art extends outward 4 inches from the wall.

52. DISTANCE ACROSS A MARSH Find the distance AB across the marsh shown in the accompanying figure.

4 in.

38° 6 ft

A

52° B

C 31 m

53. WIDTH OF A RAMP A skateboarder wishes to build a jump ramp that is inclined at a 19 angle and that has a maximum height of 32.0 inches. Find the horizontal width x of the ramp.

56. HEIGHT OF THE EIFFEL TOWER The angle of elevation from a point 116 meters from the base of the Eiffel Tower to the top of the tower is 68.9 . Find the approximate height of the tower. 57. DISTANCE OF A DESCENT An airplane traveling at 240 mph is descending at an angle of depression of 6 . How many miles will the plane descend in 4 minutes?

32.0 in. 19° x

54. TIME OF CLOSEST APPROACH At 3:00 P.M., a boat is 12.5 miles due west of a radar station and traveling at 11 mph in a direction that is 57.3 south of an east-west line. At what time will the boat be closest to the radar station?

58. TIME OF A DESCENT A submarine traveling at 9.0 mph is descending at an angle of depression of 5 . How many minutes, to the nearest tenth, does it take the submarine to reach a depth of 80 feet? 59. HEIGHT OF A BUILDING A surveyor determines that the angle of elevation from a transit to the top of a building is 27.8 . The transit is positioned 5.5 feet above ground level and 131 feet from the building. Find the height of the building to the nearest tenth of a foot.

12.5 mi 57.3°

Transit 27.8°

131 ft 5.5 ft

144

Chapter 2

Trigonometric Functions

60. WIDTH OF A LAKE The angle of depression to one side of a lake, measured from a balloon 2500 feet above the lake as shown in the accompanying figure, is 43 . The angle of depression to the opposite side of the lake is 27 . Find the width of the lake.

Venus

r

46.5 Earth

43°

27°

2500 ft A

B

61. ASTRONOMY The moon Europa rotates in a nearly circular orbit around Jupiter. The orbital radius of Europa is approximately 670,900 kilometers. During a revolution of Europa around Jupiter, an astronomer found that the maximum value of the angle formed by Europa, Earth, and Jupiter was 0.056 . Find the distance d between Earth and Jupiter at the time the astronomer found the maximum value of . Round to the nearest million kilometers.

Sun

149,000,000 km

63. AREA OF AN ISOSCELES TRIANGLE Consider the following isosceles triangle. The length of each of the two equal sides of the triangle is a, and each of the base angles has a measure of . Verify that the area of the triangle is A 苷 a2 sin cos .

a

a

θ

θ b

64. AREA OF A HEXAGON Find the area of the hexagon. (Hint: The area consists of six isosceles triangles. Use the formula from Exercise 63 to compute the area of one of the triangles and multiply by 6.)

4 in.

4 in. 60° 60° 4 in.

Europa

r = 670,900 km

Earth

= 0.056 d Jupiter

65. HEIGHT OF A PYRAMID The angle of elevation to the top of the Egyptian pyramid of Cheops is 36.4 , measured from a point 350 feet from the base of the pyramid. The angle of elevation from the base of a face of the pyramid is 51.9 . Find the height of the Cheops pyramid.

Not drawn to scale.

62. ASTRONOMY Venus rotates in a nearly circular orbit around the sun. The largest angle formed by Venus, Earth, and the sun is 46.5 . The distance from Earth to the sun is approximately 149,000,000 kilometers. See the following figure. What is the orbital radius r of Venus? Round to the nearest million kilometers.

51.9° 36.4° 350 ft

2.2 66. HEIGHT OF A BUILDING Two buildings are 240 feet apart. The angle of elevation from the top of the shorter building to the top of the other building is 22 . If the shorter building is 80 feet high, how high is the taller building? 67. HEIGHT OF THE WASHINGTON MONUMENT From a point A on a line from the base of the Washington Monument, the angle of elevation to the top of the monument is 42.0 . From a point 100 feet away from A and on the same line, the angle to the top is 37.8 . Find the height, to the nearest foot, of the Washington Monument.

Right Triangle Trigonometry

connected by a skybridge at the forty-first floor. Note the information given in the accompanying figure. a. Determine the height of the skybridge. b. Determine the length of the skybridge.

C

37.8°

42.0°

D

AB = 412 feet CAB = 53.6˚ AB is at ground level CAD = 15.5˚

B

A

145

A

100 ft

68. HEIGHT OF A TOWER The angle of elevation from a point A to the top of a tower is 32.1 . From point B, which is on the same line but 55.5 feet closer to the tower, the angle of elevation is 36.7 . Find the height of the tower.

36.7°

32.1° B

70. AN EIFFEL TOWER REPLICA Use the information in the accompanying figure to estimate the height of the Eiffel Tower replica that stands in front of the Paris Las Vegas Hotel in Las Vegas, Nevada.

A

55.5 ft

Street level

46.3° 65.5°

69. THE PETRONAS TOWERS The Petronas Towers in Kuala Lumpur, Malaysia, are the world’s tallest twin towers. Each tower is 1483 feet in height. The towers are

235 feet

146

Chapter 2

Trigonometric Functions

Connecting Concepts 71. A circle is inscribed in a regular hexagon with each side 6.0 meters long. Find the radius of the circle.

3 ft

72. Show that the area A of the triangle given in the figure is 1 A 苷 ab sin . 2

θ

c

a θ

3 ft

b

73. FIND A MAXIMUM LENGTH Find the length of the longest piece of wood that can be slid around the corner of the hallway in the figure in the column at right. Round to the nearest tenth of a foot.

74. FIND A MAXIMUM LENGTH In Exercise 73, suppose that the hall is 8 feet high. Find the length of the longest piece of wood that can be taken around the corner. Round to the nearest tenth of a foot.

Projects 1. a. PERIMETER OF A REGULAR n-GON Show that the perimeter P of a regular n-sided polygon (n-gon) inscribed in a circle of radius 1 is P 苷 2n sin b.

180 . n

Let Pn denote the perimeter of a regular n-gon inscribed in a circle of radius 1. Use the result from a. to complete the following table. n

10

50

100

1000

2. a. AREA OF A REGULAR n-GON Show that the area A of a regular n-gon inscribed in a circle of radius 1 is

10,000

A苷 b.

360 n sin . 2 n

Let An denote the area of a regular n-gon inscribed in a circle of radius 1. Use the result from a. to complete the following table. n

10

50

100

1000

10,000

Pn

An

Write a few sentences explaining why Pn approaches 2 as n increases without bound.

Write a few sentences explaining why An approaches as n increases without bound.

2.3

Trigonometric Functions of Any Angle

Section 2.3 ■

■

■

■

147

Trigonometric Functions of Any Angle

P R E PA R E F O R T H I S S E C T I O N

Trigonometric Functions of Any Angle Trigonometric Functions of Quadrantal Angles Signs of Trigonometric Functions The Reference Angle

Prepare for this section by completing the following exercises. The answers can be found on page A9.

PS1. Find the reciprocal of

3 . 4

PS2. Find the reciprocal of

PS3. Evaluate: 兩120 180兩 [1.1] PS5. Simplify:

PS4. Simplify: 2p

p 3 p 2 2

2兹 5 . 5

9p 5

PS6. Simplify: 兹共3兲2 共5兲2 [1.2]

Trigonometric Functions of Any Angle y

P(x, y) r

y

θ

x

x

O

The applications of trigonometry would be quite limited if all angles had to be acute angles. Fortunately, this is not the case. In this section we extend the definition of a trigonometric function to include any angle. Consider angle u in Figure 2.41 in standard position and a point P共x, y兲 on the terminal side of the angle. We define the trigonometric functions of any angle according to the following definitions.

Figure 2.41

Definitions of the Trigonometric Functions of Any Angle Let P共x, y兲 be any point, except the origin, on the terminal side of an angle u in standard position. Let r 苷 d共O, P兲, the distance from the origin to P. The six trigonometric functions of u are y r r csc u 苷 , y

sin u 苷

x r r sec u 苷 , x

cos u 苷 y苷0

y , x苷0 x x cot u 苷 , y 苷 0 y

tan u 苷 x苷0

where r 苷兹x 2 y 2.

y

P(a′, b′) r′ P(a, b) r

b′

b

θ

x

a a′

Figure 2.42

The value of a trigonometric function is independent of the point chosen on the terminal side of the angle. Consider any two points on the terminal side of an angle u in standard position, as shown in Figure 2.42. The right triangles formed are similar triangles, so the ratios of the corresponding sides are equal. Thus, b b b b b for example, 苷 . Because tan u 苷苷 , we have tan u 苷 . Therefore, the a a a a a value of the tangent function is independent of the point chosen on the terminal side of the angle. By a similar argument, we can show that the value of any trigonometric function is independent of the point chosen on the terminal side of the angle.

148

Chapter 2

Trigonometric Functions Any point in a rectangular coordinate system (except the origin) can determine an angle in standard position. For example, P共4, 3兲 in Figure 2.43 is a point in the second quadrant and determines an angle u in standard position with r 苷兹共4兲2 32 苷 5. y

P(−4, 3) 5 θ

x

Figure 2.43

The values of the trigonometric functions of u as shown in Figure 2.43 are sin u 苷

3 5

cos u 苷

4 4 苷 5 5

tan u 苷

3 3 苷 4 4

csc u 苷

5 3

sec u 苷

5 5 苷 4 4

cot u 苷

4 4 苷 3 3

EXAMPLE 1

»

Evaluate Trigonometric Functions

Find the exact value of each of the six trigonometric functions of an angle u in standard position whose terminal side contains the point P共3, 2兲.

Solution The angle is sketched in Figure 2.44. Find r by using the equation r 苷兹x 2 y 2, where x 苷 3 and y 苷 2.

y

r 苷兹共3兲2 共2兲2 苷兹9 4 苷兹13

4

Now use the definitions of the trigonometric functions.

2 θ

−4 P(−3, −2)

−2

2 r

4 x

sin u 苷

−2

csc u 苷 Figure 2.44

2 兹13

苷

2 兹13 13

兹13 兹13 苷 2 2

cos u 苷 sec u 苷

3 兹13

苷

3 兹13 13

兹13 兹13 苷 3 3

tan u 苷

2 2 苷 3 3

cot u 苷

3 3 苷 2 2

»

Try Exercise 6, page 153

Trigonometric Functions of Quadrantal Angles Recall that a quadrantal angle is an angle whose terminal side coincides with the x- or y-axis. The value of a trigonometric function of a quadrantal angle can be found by choosing any point on the terminal side of the angle and then applying the definition of that trigonometric function.

2.3

149

Trigonometric Functions of Any Angle

The terminal side of 0 coincides with the positive x-axis. Let P共x, 0兲, x 0, be a point on the x-axis, as shown in Figure 2.45. Then y 苷 0 and r 苷 x. The values of the six trigonometric functions of 0 are

y

P(x, 0) x

sin 0 苷 Figure 2.45

0 苷0 r

csc 0 is undefined.

QUESTION

cos 0 苷

x x 苷苷1 r x

tan 0 苷

0 苷0 x

sec 0 苷

r x 苷苷1 x x

cot 0 is undefined.

Why are csc 0° and cot 0° undefined?

In like manner, the values of the trigonometric functions of the other quadrantal angles can be found. The results are shown in Table 2.4. Table 2.4

y

Values of Trigonometric Functions of Quadrantal Angles sin

cos

tan

csc

undefined

0°

0

1

0

90°

1

0

undefined

180°

0

1

270°

1

0

sec

1

1

cot

undefined

undefined

0

0

undefined

1

undefined

undefined

1

undefined

0

θ

x

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies. For example, if u is an angle whose terminal side lies in Quadrant III and P共x, y兲 is on the terminal side of u, then both x and y are negay x y tive, and therefore and are positive. See Figure 2.46. Because tan u 苷 x y x x and cot u 苷 , the values of the tangent and cotangent functions are positive for y any Quadrant III angle. The values of the other four trigonometric functions of any Quadrant III angle are all negative. Table 2.5 lists the signs of the six trigonometric functions in each quadrant. Figure 2.47 is a graphical display of the contents of Table 2.5.

P(x, y)

Figure 2.46 y

Sine and cosecant positive

Signs of Trigonometric Functions

All functions positive

x

Tangent and cotangent positive

Cosine and secant positive

Figure 2.47

Table 2.5

Signs of the Trigonometric Functions

Sign of

I

Terminal Side of in Quadrant II III

IV

sin u and csc u

positive

positive

negative

negative

cos u and sec u

positive

negative

negative

positive

tan u and cot u

positive

negative

positive

negative

ANSWER

P共x, 0兲 is a point on the terminal side of a 0 angle in standard position. Thus x r csc 0 苷 , which is undefined. Similarly, cot 0 苷 , which is undefined. 0 0

150

Chapter 2

Trigonometric Functions In the next example we are asked to evaluate two trigonometric functions of the angle u. A key step is to use our knowledge about trigonometric functions and their signs to determine that u is a Quadrant IV angle.

Given tan u 苷

7 and sin u 0, find cos u and csc u. 5

Evaluate Trigonometric Functions

The terminal side of angle u must lie in Quadrant IV; that is the only quadrant in which sin u and tan u are both negative. Because

2

−2

»

Solution

y

−2

EXAMPLE 2

θ

6

8

x

tan u 苷

−4 −6

74 (5, −7)

−8

7 y 苷 5 x

(1)

and the terminal side of u is in Quadrant IV, we know that y must be negative and x must be positive. Thus Equation (1) is true for y 苷 7 and x 苷 5. Now r 苷兹52 共7兲2 苷兹74. See Figure 2.48. Hence

Figure 2.48

cos u 苷

x 5 5 兹74 苷苷 r 74 兹74

csc u 苷

r 兹74 兹74 苷苷 y 7 7

and

take note In this section r is a distance and hence nonnegative.

»

Try Exercise 30, page 153

The Reference Angle We will often find it convenient to evaluate trigonometric functions by making use of the concept of a reference angle.

take note The reference angle is a very important concept that will be

Definition of a Reference Angle Given ⬔u in standard position, its reference angle u is the acute angle formed by the terminal side of ⬔u and the x-axis.

used time and time again in the remaining trigonometry sections. Figure 2.49 shows ⬔u and its reference angle u for four cases. In every case the reference angle u is formed by the terminal side of ⬔u and the x-axis (never the y-axis). The process of determining the measure of ⬔u varies according to which quadrant contains the terminal side of ⬔u.

2.3 y

y

θ' = θθ

y

θ

θ'

y

θ

x

x

θ

x

θ'

θ'

If 180° < θ < 270°, then θ ' = θ − 180°.

If 90° < θ < 180°, then θ ' = 180° − θ .

If 0° < θ < 90°, then θ ' = θ .

151

Trigonometric Functions of Any Angle

If 270° < θ < 360°, then θ ' = 360° − θ .

Figure 2.49

»

EXAMPLE 3

Find the Measure of a Reference Angle

Find the measure of the reference angle u for each angle. u 苷 120

a.

b.

u 苷 345

c.

u苷

7p 4

u苷

d.

13p 6

Solution For any angle in standard position, the measure of its reference angle is the measure of the acute angle formed by its terminal side and the x-axis. a.

b.

y

y θ = 345°

θ = 120°

θ '= 60°

θ '= 15°

x

x

Integrating Technology A TI-83/TI-83 Plus/TI-84 Plus graphing calculator program is available to compute the measure of the reference angle for a given angle. This program, REFANG, can be found on our website at college.hmco.com/info/ aufmannCAT

u 苷 180 120 苷 60 c.

u 苷 360 345 苷 15 d.

y

y

θ = 7π 4

θ' = π 6 θ' = π 4

u 苷 2p

»

7p p 苷 4 4

Try Exercise 38, page 153

x

x θ = 13π 6

u 苷

13p p 2p 苷 6 6

x

152

Chapter 2

Trigonometric Functions The following theorem states an important relationship that exists between sin u and sin u , where u is the reference angle for angle u.

take note The Reference Angle Theorem is also valid if the sine function is

Reference Angle Theorem To evaluate sin u, determine sin u . Then use either sin u or its opposite as the answer, depending on which has the correct sign.

replaced by any other trigonometric function.

In the following example, we illustrate how to evaluate a trigonometric function of u by first evaluating the trigonometric function of u . EXAMPLE 4

»

Use the Reference Angle Theorem to Evaluate Trigonometric Functions

Determine the exact value of each function. sin 210

a.

b.

cos 405

c.

tan

5p 3

Solution We know that sin 210 is negative (the sign chart is given in Table 2.5 on page 149). The reference angle for u 苷 210 is u 苷 30 . By the Reference Angle Theorem, we know that sin 210 equals either

a.

sin 30 苷 Thus sin 210 苷

1 2

or

sin 30 苷

1 2

1 . 2

Because u 苷 405 is a Quadrant I angle, we know that cos 405 0. The reference angle for u 苷 405 is u 苷 45 . By the reference angle theorem, cos 405 equals either

b.

cos 45 苷 Thus cos 405 苷

兹2 2

or

cos 45 苷

兹2 2

兹2 . 2

5p 5p is a Quadrant IV angle, tan 0. The 3 3 5p p 5p reference angle for u 苷 is u 苷 . Hence tan equals either 3 3 3 p p tan 苷兹3 or tan 苷兹3 3 3 Because u 苷

c.

Thus tan

»

5p 苷兹3. 3

Try Exercise 50, page 154

2.3

153

Trigonometric Functions of Any Angle

Topics for Discussion 1. Is every reference angle an acute angle? Explain. 2. If u is the reference angle for the angle u, then sin u 苷 sin u . Do you agree? Explain. 3. If sin u 0 and cos u 0, then the terminal side of the angle u lies in which quadrant? 4. Explain how to find the measure of the reference angle u for the angle u 苷

19 p. 5

Exercise Set 2.3 In Exercises 1 to 8, find the value of each of the six trigonometric functions for the angle whose terminal side passes through the given point.

1. P共2, 3兲

2. P共3, 7兲

3. P共2, 3兲

4. P共3, 5兲

5. P共8, 5兲

6. P共6, 9兲

7. P共5, 0兲

8. P共0, 2兲

12. sec 90° 15 cos

p 2

18 cot p

10. cos 270°

11. tan 180°

13. csc 90°

14. cot 90°

16. sin

30. sec u 苷

3p 2

17. tan

p 19. sin 2

p 2

21. sin u 0,

cos u 0

22. tan u 0,

sin u 0

23. cos u 0,

tan u 0

24. sin u 0,

cos u 0

25. sin u 0,

cos u 0

26. tan u 0,

cos u 0

1 and cos u 0; find tan u. 2

32. tan u 苷 1 and sin u 0; find cos u. 1 and tan u 苷兹 3; find csc u. 2

33. cos u 苷

34. tan u 苷 1 and sin u 苷

35. cos u 苷

20. cos p

In Exercises 21 to 26, let be an angle in standard position. State the quadrant in which the terminal side of lies.

p u p ; find cot u. 2

2兹 3 3p , u 2p ; find sin u. 3 2

31. sin u 苷

In Exercises 9 to 20, evaluate the trignometric function of the quadrantal angle, or state that the function is undefined.

9. sin 180°

29. csc u 苷兹 2,

36. sec u 苷

兹2 ; find sec u. 2

1 兹3 and sin u 苷 ; find cot u. 2 2

1 2兹3 and sin u 苷 ; find cot u. 3 2

In Exercises 37 to 48, find the measure of the reference angle for the given angle .

37. u 苷 160

38. u 苷 255

40. u 苷 48

41. u 苷

11 p 5

42. u 苷 6

44. u 苷

18 p 7

45. u 苷 1406

In Exercises 27 to 36, find the exact value of each expression.

27. sin u 苷

1 , 180 u 270 ; find tan u. 2

28. cot u 苷 1, 90 u 180 ; find cos u.

43. u 苷

8 3

46. u 苷 840

47. u 苷 475

39. u 苷 351

48. u 苷 650

154

Chapter 2

Trigonometric Functions

In Exercises 49 to 60, use the Reference Angle Theorem to find the exact value of each trigonometric function.

In Exercises 73 to 80, find (without using a calculator) the exact value of each expression.

49. sin 225

50. cos 300

73. sin 210 cos 330 tan 330

52. sec 150

53. csc

55. cos

17p 4

冉冊冉冊 4 p 3

56. tan

58. csc 共510 兲

p 3

59. cot 540

51. tan 405 54. cot

冉冊 7 p 6

57. sec 765

74. tan 225 sin 240 cos 60 75. sin2 30 cos2 30 76. cos p sin

60. cos 570 77. sin

In Exercises 61 to 72, use a calculator to approximate the given trigonometric function to six significant digits.

61. sin 127

62. sin 共257 兲

63. cos 共116 兲

64. cot 398

65. sec 578

66. sec 740

冉冊

67. sin

p 5

70. tan 共4.12兲

68. cos

3p 7

69. csc

71. sec 共4.45兲

9p 5

78. cos

11p 7p tan 4 6

冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊 3p tan 2

p 4

cos

p 3

7p 4p 7p tan cos 4 3 6

79. sin2

5p 5p cos2 4 4

80. tan2

7p 7p sec2 4 4

72. csc 0.34

Connecting Concepts In Exercises 81 to 86, find two values of , 0 360, that satisfy the given trigonometric equation.

81. sin u 苷

1 2

83. cos u 苷

82. tan u 苷兹3 兹3 2

85. csc u 苷兹2

84. tan u 苷 1

cos2 sin2

1 2

87. tan u 苷 1

88. cos u 苷

兹3 89. tan u 苷 3

2兹 3 90. sec u 苷 3

兹3 2

冉冊冉冊 x r

86. cot u 苷 1

In Exercises 87 to 92, find two values of , 0 2 , that satisfy the given trigonometric equation.

91. sin u 苷

If P(x, y) is a point on the terminal side of an acute angle y in standard position and r 兹x2 y2 , then sin and r x cos . Using these definitions, we find that r

92. cos u 苷

1 2

2

y r

2

x2 y2 x2 y2 . 2 2 r r r2

r2 1 r2

Hence cos2 sin2 1 for all acute angles . This important identity is actually true for all angles . We will show this later. In the meantime, use the definitions of the trigonometric functions to prove that the equations are identities for the acute angle in Exercises 93 to 96.

93. 1 tan2 u 苷 sec 2 u

94. cot 2 u 1 苷 csc 2 u

95. cos 共90 u兲苷 sin u

96. sin 共90 u兲苷 cos u

2.4

Trigonometric Functions of Real Numbers

155

Projects 1.

FIND SUMS OR PRODUCTS Determine the following sums or products. Do not use a calculator. (Hint: The Reference Angle Theorem may be helpful.) Explain to a classmate how you know you are correct. a. cos 0 cos 1 cos 2 cos 178 cos 179 cos 180 b. sin 0 sin 1 sin 2 sin 358 sin 359 sin 360 c. cot 1 cot 2 cot 3 cot 177 cot 178 cot 179 d. 共cos 1 兲共cos 2 兲共cos 3 兲共cos 177 兲共cos 178 兲共cos 179 兲 e. cos2 1 cos2 2 cos2 3 cos2 357 cos2 358 cos2 359

Section 2.4 ■ ■

■

■

The Wrapping Function Trigonometric Functions of Real Numbers Properties of Trigonometric Functions of Real Numbers Trigonometric Identities

Trigonometric Functions of Real Numbers P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A10.

PS1. Determine whether the point (0, 1) is a point on the circle defined by x 2 y 2 苷 1. [1.1/1.2] PS2. Determine whether the point x 2 y 2 苷 1. [1.1/1.2] PS3. Determine whether the point x y 苷 1. [1.1/1.2] 2

2

冉冊 1 兹3 , 2 2

冉

兹2 兹3 , 2 2

is a point on the circle defined by

冊

is a point on the circle defined by

PS4. Determine the circumference of a circle with a radius of 1. [2.1] PS5. Determine whether f 共x兲苷 x 2 3 is an even function, an odd function, or neither. [1.4] PS6. Determine whether f共x兲苷 x 3 x 2 is an even function, an odd function, or neither. [1.4]

The Wrapping Function In the seventeenth century, applications of trigonometry were extended to problems in physics and engineering. These kinds of problems required trigonometric functions whose domains were sets of real numbers rather than sets of angles. This extension of the definitions of trigonometric functions to include real numbers was accomplished by using a correspondence between an angle and the length of an arc on a unit circle.

156

Chapter 2

Trigonometric Functions Consider a circle given by the equation x 2 y 2 苷 1, called a unit circle, and a vertical coordinate line l tangent to the unit circle at 共1, 0兲. We define a function W that pairs a real number t on the coordinate line with a point P共x, y兲 on the unit circle. This function W is called the wrapping function because it is analogous to wrapping a line around a circle. As shown in Figure 2.50, the positive part of the coordinate line is wrapped around the unit circle in a counterclockwise direction. The negative part of the coordinate line is wrapped around the circle in a clockwise direction. The wrapping function is defined by the equation W共t兲苷 P共x, y兲, where t is a real number y and P共x, y兲 is the point on the unit circle that P(x, y) corresponds to t. Through the wrapping function, each ២ that subreal number t defines an arc AP t t r=1 tends a central angle with a measure of ២ θ radians. The length of the arc AP is t (see Figx A(1, 0) r=1 ure 2.51). From the equation s 苷 r for the arc length of a circle, we have (with t 苷 s) t 苷 r. For a unit circle, r 苷 1, and the equation bel comes t 苷 . Thus, on a unit circle, the measure of a central angle and the length of its arc can be represented by the same real number t. Figure 2.51

y l

P1(x1, y1) t1

x

A(1, 0)

P2(x2, y2)

− t2

Figure 2.50

EXAMPLE 1

»

Evaluate the Wrapping Function

冉冊

p . 3

Evaluate W

Solution y P

( 12 ,

3 2

) π 3

θ =

π 3

p on line l is shown in Figure 2.52. From the wrapping 3 p ២ subtends function, W is the point P on the unit circle for which arc AP 3 p an angle , the measure of which is radians. The coordinates of P can be 3 determined from the definitions of cos and sin given in Section 2.3 and from Table 2.2, page 137. The point

π 3

A(1, 0)

x

冉冊

x r p x cos 苷苷x 3 1 1 苷x 2 cos 苷

l

Figure 2.52

y r p y sin 苷苷y 3 1 sin 苷

兹3 苷y 2

From these equations, x 苷

»

Try Exercise 10, page 166

•

;r1 3

• cos

1

兹3 ; sin 3 2 3 2

冉冊冉冊

1 p 兹3 and y 苷 . Therefore, W 2 2 3

苷

1 兹3 , . 2 2

2.4

Trigonometric Functions of Real Numbers

冉冊

y

p , recall that the circumference of a unit circle is 2. One2 1 p p fourth the circumference is 共2p兲苷 (see Figure 2.53). Thus W 苷 P共0, 1兲. 4 2 2 Note from the last two examples that for the given real number t, cos t 苷 x and sin t 苷 y. That is, for a real number t and W共t兲苷 P共x, y兲, the value of the cosine of t is the x-coordinate of P, and the value of the sine of t is the y-coordinate of P. See Figure 2.54. To determine W

P(0, 1)

π 2

π 2

π 2

θ =

A(1, 0)

x

l

冉冊

冉冊

What is the point defined by W

QUESTION

Figure 2.53

157

p ? 4

Trigonometric Functions of Real Numbers

y

The following definition makes use of the wrapping function W共t兲 to define trigonometric functions of real numbers.

P(x, y) = (cos t, sin t) t A(1, 0)

x

Definitions of the Trigonometric Functions of Real Numbers x 苷 cos t, y 苷 sin t

Let W be the wrapping function, t be a real number, and W共t兲苷 P共x, y兲. Then

Figure 2.54

sin t 苷 y csc t 苷

y

P(x', y') r 1 θ

y

P(x, y) t A(1, 0)

y'

x

1 , y

cos t 苷 x y苷0

sec t 苷

x苷0

Trigonometric functions of real numbers are frequently called circular functions to distinguish them from trigonometric functions of angles. The trigonometric functions of real numbers (or circular functions) look remarkably like the trigonometric functions defined in the last section. The difference between the two is that of domain: In one case, the domains are sets of real numbers; in the other case, the domains are sets of angle measurements. However, there are similarities between the two functions. Consider an angle (measured in radians) in standard position, as shown in Figure 2.55. Let P共x, y兲 and P 共x , y 兲 be two points on the terminal side of , where x 2 y 2 苷 1 and 共x 兲2 共y 兲2 苷 r 2. Let t be the length of the arc from A共1, 0兲 to P共x, y兲. Then sin 苷

Figure 2.55

1 , x

y , x苷0 x x cot t 苷 , y 苷 0 y tan t 苷

y y 苷苷 sin t r 1

Thus the value of the sine function of , measured in radians, is equal to the value of the sine of the real number t. Similar arguments can be given to show cor-

ANSWER

冉

兹2 兹2 , 2 2

冊

158

Chapter 2

Trigonometric Functions responding results for the other five trigonometric functions. With this in mind, we can assert that the value of a trigonometric function at the real number t is its value at an angle of t radians.

EXAMPLE 2

»

Evaluate Trigonometric Functions of Real Numbers

Find the exact value of each function. cos

a.

4

冉冊

sin

b.

7 6

c.

冉冊

tan

5 4

d.

sec

5 3

Solution The value of a trigonometric function at the real number t is its value at an angle of t radians. Using Table 2.2 on page 137, we have

兹2 苷 4 2

a.

cos

b.

sin

c.

tan

d.

sec

冉冊冉冊 7 6

苷 sin

1 苷 6 2

5 4

苷 tan

苷 1 4

5 苷 sec 苷2 3 3

»

7 is 6 6 and sin t 0 in Quadrant II.

• Reference angle for

5 is 4 4 and tan t 0 in Quadrant II.

• Reference angle for

5 is 3 3 and sec t 0 in Quadrant IV.

• Reference angle for

Try Exercise 16, page 166

In Example 3 we evaluate a trigonometric function of a real number to solve an application.

EXAMPLE 3

»

Determine a Height as a Function of Time

The Millennium Wheel, in London, is the world’s largest Ferris wheel. It has a diameter of 450 feet. When the Millennium Wheel is in uniform motion, it completes one revolution every 30 minutes. The height h, in feet above the Thames River, of a person riding on the Millennium Wheel can be estimated by

冉冊

h共t兲苷 255 225 cos

p t 15

where t is the time in minutes since the person started the ride.

2.4

159

Trigonometric Functions of Real Numbers

a.

How high is the person at the start of the ride 共t 苷 0兲?

b.

How high is the person after 18.0 minutes? Round to the nearest foot.

Solution

冉冊

h共0兲苷 255 225 cos

a.

苷 255 225 苷 30

0 15

冉

b. h共18.0兲苷 255 225 cos

⬇ 255 共182兲苷 437

At the start of the ride, the person is 30 feet above the Thames.

18.0 15

冊

After 18.0 minutes, the person is about 437 feet above the Thames.

»

Try Exercise 82, page 167

The Millennium Wheel, on the banks of the Thames River, London.

TO REVIEW

Domain and Range See page 32.

Properties of Trigonometric Functions of Real Numbers The domain and range of the trigonometric functions of real numbers can be found from the definitions of these functions. If t is any real number and P共x, y兲 is the point corresponding to W共t兲, then by definition cos t 苷 x and sin t 苷 y. Thus the domain of the sine and cosine functions is the set of real numbers. Because the radius of the unit circle is 1, we have 1 x 1

and

1 y 1

Therefore, with x 苷 cos t and y 苷 sin t, we have 1 cos t 1

and

1 sin t 1

The range of the cosine and sine functions is 关1, 1兴. Using the definitions of tangent and secant, tan t 苷

y x

and

sec t 苷

1 x

The domain of the tangent function is all real numbers t except those for which the 3 x-coordinate of W共t兲 is zero. The x-coordinate is zero when t 苷 , t 苷 , 2 2 5p 共2n 1兲 t 苷 , and in general when t 苷 , where n is an integer. 2 2 Thus the domain of the tangent function is the set of all real numbers t except 共2n 1兲 t苷 , where n is an integer. The range of the tangent function is all 2 real numbers. Similar methods can be used to find the domain and range of the cotangent, secant, and cosecant functions. The results are summarized in Table 2.6 on the next page.

160

Chapter 2

Trigonometric Functions Table 2.6 Domain and Range of the Trigonometric Functions of Real Numbers (n is an integer) Function

兵t 兩 t 其

兵 y 兩 1 y 1其

y 苷 cos t

兵t 兩 t 其

兵 y 兩 1 y 1其

l t

y 苷 csc t

P1(x, y)

y 苷 sec t A(1, 0) x P2(x, −y)

Figure 2.56

−t

y 苷 cot t

Odd and Even Functions See page 56.

再

t 兩 t , t 苷

冎

共2n 1兲p 2

兵t 兩 t , t 苷 np 其

再

t 兩 t , t 苷

冎

共2n 1兲p 2

兵t 兩 t , t 苷 np 其

兵 y 兩 y 其兵 y 兩 y 1, y 1其兵 y 兩 y 1, y 1其兵 y 兩 y 其

Consider the points t and t on the coordinate line l tangent to the unit circle at the point 共1, 0兲. The points W共t兲 and W共t兲 are symmetric with respect to the x-axis. Therefore, if P 1共x, y兲 are the coordinates of W共t兲, then P 2共x, y兲 are the coordinates of W共t兲. See Figure 2.56. From the definitions of the trigonometric functions, we have sin t 苷 y

TO REVIEW

Range

y 苷 sin t

y 苷 tan t

y

Domain

and sin共t兲苷 y

and

cos t 苷 x

and cos共t兲苷 x

Substituting sin t for y and cos t for x yields sin共t兲苷 sin t

and

cos共t兲苷 cos t

Thus the sine is an odd function and the cosine is an even function. Because 1 1 csc t 苷 and sec t 苷 , it follows that sin t cos t csc共t兲苷 csc t

and

sec共t兲苷 sec t

These equations show that the cosecant is an odd function and the secant is an even function. y From the definition of the tangent function, we have tan t 苷 and x y y tan共t兲苷 . Substituting tan t for yields tan共t兲苷 tan t. Because x x 1 cot t 苷 , it follows that cot共t兲苷 cot t. Thus the tangent and cotangent tan t functions are odd functions.

Even and Odd Trigonometric Functions The odd trigonometric functions are y 苷 sin t, y 苷 csc t, y 苷 tan t, and y 苷 cot t. The even trigonometric functions are y 苷 cos t and y 苷 sec t. Thus for all t in their domain, sin共t兲苷 sin t csc共t兲苷 csc t

cos共t兲苷 cos t sec共t兲苷 sec t

tan共t兲苷 tan t cot共t兲苷 cot t

2.4 EXAMPLE 4

Trigonometric Functions of Real Numbers

»

161

Determine Whether a Function Is Even, Odd, or Neither

Is f共x兲苷 x tan x an even function, an odd function, or neither?

Solution Find f共x兲 and compare it to f共x兲. f共x兲苷共x兲 tan共x兲苷 x tan x 苷共x tan x兲苷 f共x兲

• tan(x) tan x

The function f共x兲苷 x tan x is an odd function.

»

Try Exercise 44, page 166

We encounter many recurring patterns in everyday life. For instance, the time of day repeats every 24 hours. If f共t兲 represents the present time of day, then 24 hours later the time of day will be exactly the same. Using mathematical notation, we can express this concept as f共t 24兲苷 f共t兲 The function f is said to be periodic in that it repeats itself over and over. The period of f is 24 hours, the time it takes to complete one full cycle.

Definition of a Periodic Function A function f is periodic if there exists a positive constant p such that f共t p兲苷 f共t兲 for all t in the domain of f. The smallest such positive number p for which f is periodic is called the period of f.

The unit circle can be used to show that the sine and cosine functions are periodic functions. First note that the circumference of the unit circle is 2p. Thus, if we start at any point P共x, y兲苷 P共cos t, sin t兲 on the unit circle and travel a distance of 2p units around the circumference, we will be back at point P. Hence 共cos共t 2p兲, sin共t 2p兲兲苷共cos t, sin t兲. Equating the first componenets gives us cos共t 2p兲苷 cos t and equating the second components yields sin共t 2p兲苷 sin t. The following equations illustrate that the secant and cosecant functions are also periodic functions. 1 1 苷苷 sec t cos共t 2p兲 cos t 1 1 csc共t 2p兲苷苷苷 csc t sin共t 2p兲 sin t

sec共t 2p兲苷

162

Chapter 2

Trigonometric Functions Period of the Sine, Cosine, Secant, and Cosecant Functions The sine, cosine, secant, and cosecant functions are periodic functions with a period of 2p. sin共t 2p兲苷 sin t sec共t 2p兲苷 sec t

cos共t 2p兲苷 cos t csc共t 2p兲苷 csc t

y P(x, y) t+π

t

A(1, 0)

Q(−x, −y)

x

Although it is true that tan共t 2p兲苷 tan t, the period of the tangent function is not 2p. Recall that the period of a function is the smallest value of p for which f共t p兲苷 f共t兲. Examine Figure 2.57, which shows that if you start at any point P共x, y兲 on the unit circle and travel a distance of p units around the circumference, you will arrive at the point 共x, y兲. By definition, tan t 苷

Figure 2.57

y x

and

tan共t p兲苷

y y 苷苷 tan t x x

Thus we know that tan共t p兲苷 tan t for all t. A similar argument can be used to show that cot共t p兲苷 cot t for all t.

Period of the Tangent and Cotangent Functions The tangent and cotangent functions are periodic functions with a period of p. tan共t p兲苷 tan t

and

cot共t p兲苷 cot t

The following theorem illustrates the repetitive nature of each of the trigonometric functions of a real number.

Periodic Properties of the Trigonometric Functions of a Real Number For any real number t and integer k, sin共t 2kp兲苷 sin t sec共t 2kp兲苷 sec t tan共t kp兲苷 tan t

cos共t 2kp兲苷 cos t csc共t 2kp兲苷 csc t cot共t kp兲苷 cot t

2.4

Trigonometric Functions of Real Numbers

163

Trigonometric Identities Recall that any equation that is true for every number in the domain of the equation is an identity. The statement csc t 苷

1 , sin t

sin t 苷 0

is an identity because the two expressions produce the same result for all values of t for which both functions are defined. The ratio identities are obtained by writing the tangent and cotangent functions in terms of the sine and cosine functions. tan t 苷

y sin t 苷 x cos t

and

cot t 苷

cos t x 苷 y sin t

• x cos t and y sin t

The Pythagorean identities are based on the equation of a unit circle, x 2 y 2 苷 1, and on the definitions of the sine and cosine functions. x2 y2 苷 1 cos2 t sin2 t 苷 1

• Replace x by cos t and y by sin t.

Dividing each term of cos t sin t 苷 1 by cos2 t, we have 2

2

cos2 t sin2 t 1 苷 2 cos t cos2 t cos2 t

• cos t 0

1 tan2 t 苷 sec2 t

sin t tan t cos t

•

Dividing each term of cos2 t sin2 t 苷 1 by sin2 t, we have cos2 t sin2 t 1 2 2 苷 sin t sin t sin2 t

• sin t 0

cot 2 t 1 苷 csc2 t

•

cos t cot t sin t

Here is a summary of the Fundamental Trigonometric Identities:

Fundamental Trigonometric Identities The reciprocal identities are sin t 苷

1 csc t

cos t 苷

1 sec t

tan t 苷

tan t 苷

sin t cos t

cot t 苷

cos t sin t

1 cot t

The ratio identities are

The Pythagorean identities are cos2 t sin2 t 苷 1

1 tan2 t 苷 sec2 t

1 cot 2 t 苷 csc2 t

164

Chapter 2

Trigonometric Functions EXAMPLE 5

»

Use the Unit Circle to Verify an Identity

Use the unit circle and the definitions of the trigonometric functions to show that sin 共t 兲苷 sin t.

Solution Sketch the unit circle and let P be the point on the unit circle such that W共t兲苷 P共x, y兲, as shown in Figure 2.58. Draw a diameter from P and label the endpoint Q. For any line through the y origin, if P共x, y兲 is a point on the line, P(x, y) then Q共x, y兲 is also a point on the t+π t line. Because line segment PQ is a diameter, the length of the arc from x P to Q is . Thus the length of the arc A(1, 0) from A through P to Q is t . Therefore, W共t 兲苷 Q共x, y兲. From Q(−x, −y) the definition of sin t, we have sin t 苷 y

and

sin 共t 兲苷 y

Thus sin 共t 兲苷 sin t.

Figure 2.58

»

Try Exercise 56, page 167

Using identities and basic algebra concepts, we can rewrite trigonometric expressions in different forms.

EXAMPLE 6

»

Write the expression

take note Because 1 1 1 苷 sin2 t cos2 t sin2 t cos2 t

1 1 as a single term. 2 sin t cos2 t

Solution Express each fraction in terms of a common denominator. The common denominator is sin2 t cos2 t. 1 1 1 cos2 t 1 sin2 t 2 2 苷 2 2 2 sin t cos t sin t cos t cos t sin2 t

苷共csc2 t兲共sec2 t兲 we could have written the answer

苷

to Example 6 in terms of the cosecant and secant functions.

Simplify a Trigonometric Expression

»

cos2 t sin2 t 1 苷 sin2 t cos2 t sin2 t cos2 t

Try Exercise 72, page 167

• cos2 t sin2 t 1

2.4

Trigonometric Functions of Real Numbers

EXAMPLE 7

For

»

165

Write a Trigonometric Expression in Terms of a Given Function

t , write tan t in terms of sin t. 2

Solution Write tan t 苷

sin t . Now solve cos2 t sin2 t 苷 1 for cos t. cos t cos2 t sin2 t 苷 1 cos2 t 苷 1 sin2 t cos t 苷兹1 sin2 t

Because

t , cos t is negative. Therefore, cos t 苷兹1 sin2 t. 2

Thus tan t 苷

sin t sin t 苷 cos t 兹1 sin2 t

•

t 2

»

Try Exercise 78, page 167

Topics for Discussion 1. Is W共t兲 a number? Explain. 2. Explain why the equation cos2 t sin2 t 苷 1 is called a Pythagorean identity. 3. Is f共x兲苷 cos3 x an even function or an odd function? Explain how you made your decision. 4. Explain how to make use of a unit circle to show that sin共t兲苷 sin t.

166

Chapter 2

Trigonometric Functions

Exercise Set 2.4 In Exercises 1 to 12, evaluate W(t) for each given t.

1. t 苷

6

2. t 苷

4

4. t 苷

4 3

5. t 苷

7. t 苷

11 6

8. t 苷 0

7

䉴 10. t 苷 4

3. t 苷

5 3

11. t 苷

In Exercises 33 to 40, use the unit circle to estimate the following values to the nearest tenth.

7 6

6. t 苷

y 2

6

1

9. t 苷 2 3

12. t 苷

0.5

t

3

− 0.5

6

In Exercises 13 to 22, find the exact value of each function.

13. tan

冉冉冉冉冉

冊冊冊冊冊

11 6

15. cos

17. csc

2 3

3

3 19. sin 2 21. sec

7 6

14. cot

冉冊冉冊 2 3

16. sec

x

0.5

−0.5

4

5 6

5 Unit Circle

18. tan 共12兲

冉冊冉冊

7 20. cos 3 22. sin

5 3

33. a. sin 2

b. cos 2

34. a. sin 3

b. cos 3

35. a. sin 5.4

b. cos 5.4

36. a. sin 4.1

b. cos 4.1

37. All real numbers t between 0 and 2 for which sin t 苷 0.4 In Exercises 23 to 32, use a calculator to find an approximate value of each function. Round your answers to the nearest ten-thousandth.

23. sin 1.22

24. cos 4.22

25. csc 共1.05兲

26. sin 共0.55兲

27. tan

冉冊冉冊 11 12

29. cos

5

31. sec 1.55

28. cos

冉冊 2 5

38. All real numbers t between 0 and 2 for which cos t 苷 0.8 39. All real numbers t between 0 and 2 for which sin t 苷 0.3 40. All real numbers t between 0 and 2 for which cos t 苷 0.7 In Exercises 41 to 48, determine whether the function is even, odd, or neither.

30. csc 8.2

41. f 共x兲苷 4 sin x

42. f 共x兲苷 2 cos x

32. cot 2.11

43. G共x兲苷 sin x cos x

44. F共x兲苷 tan x sin x

2.4 45. S共x兲苷

sin x ,x苷0 x

47. v共x兲苷 2 sin x cos x

46. C共x兲苷

Trigonometric Functions of Real Numbers

cos x ,x苷0 x

48. w共x兲苷 x tan x

In Exercises 49 to 54, state the period of each function.

49. f 共t兲苷 sin t

50. f 共t兲苷 cos t

51. f 共t兲苷 tan t

52. f 共t兲苷 cot t

53. f 共t兲苷 sec t

54. f 共t兲苷 csc t

80. Write sec t in terms of tan t, t

167

3 . 2

81. PATH OF A SATELLITE A satellite is launched into space from Cape Canaveral. The directed distance, in miles, that the satellite is north or south of the equator is

冉冊

d共t兲苷 1970 cos

t 64

where t is the number of minutes since liftoff. A negative d value indicates that the satellite is south of the equator.

In Exercises 55 to 60, use the unit circle to verify each identity.

55. cos 共t兲苷 cos t

56. tan 共t 兲苷 tan t

57. cos 共t 兲苷 cos t

58. sin 共t兲苷 sin t

59. sin 共t 兲苷 sin t

60. sec 共t兲苷 sec t

1970 miles Equator

In Exercises 61 to 76, use trigonometric identities to write each expression in terms of a single trigonometric function or a constant. Answers may vary.

61. tan t cos t 63.

62. cot t sin t

csc t cot t

64.

65. 1 sec2 t 67. tan t

69.

sec2 t tan t

1 cos2 t tan2 t

tan t cot t tan t

sec t tan t

68.

csc2 t cot t cot t

70.

1 sin2 t cot 2 t

冉冊

T共t兲苷 41 cos

2

csc t sin t csc t

76. cos t共1 tan t兲 2

77. Write sin t in terms of cos t, 0 t

. 2

t 36 6

where t is the number of months after January 5. Use the formula to estimate (to the nearest tenth of a degree Fahrenheit) the average high temperature in Fairbanks for March 5 and July 20.

1 1 72. 1 sin t 1 sin t 74.

75. sin t共1 cot t兲 2

82. AVERAGE HIGH TEMPERATURE The average high temperature T, in degrees Fahrenheit, for Fairbanks, Alaska, is given by

66. 1 csc2 t

1 1 71. 1 cos t 1 cos t 73.

What distance, to the nearest 10 miles, is the satellite north of the equator 24 minutes after liftoff?

2

In Exercises 83 to 94, perform the indicated operation and simplify.

83. cos t

1 cos t

84. tan t

1 tan t

85. cot t

1 cot t

86. sin t

1 sin t

87. 共1 sin t兲2

88. 共1 cos t兲2

89. 共sin t cos t兲2

90. 共sin t cos t兲2 92. 共1 cos t兲共1 cos t兲

78. Write tan t in terms of sec t,

3 t 2. 2

91. 共1 sin t兲共1 sin t兲

79. Write csc t in terms of cot t,

t . 2

93.

sin t 1 cos t 1 cos t sin t

94.

1 sin t 1 cos t tan t sec t

168

Chapter 2

Trigonometric Functions 99. 2 sin2 t sin t 1

In Exercises 95 to 100, factor the expression.

95. cos2 t sin2 t

96. sec2 t csc2 t

97. tan2 t tan t 6

98. cos2 t 3 cos t 4

100. 4 cos2 t 4 cos t 1

Connecting Concepts In Exercises 101 to 104, use trigonometric identities to find the value of the function.

In Exercises 105 to 108, simplify the first expression to the second expression.

, find cos t. 2

105.

sin2 t cos2 t ; csc2 t sin2 t

102. Given cos t 苷

1 3 , t 2, find sin t. 2 2

106.

sin2 t cos2 t ; sec2 t cos2 t

103. Given sin t 苷

1 , t , find tan t. 2 2

101. Given csc t 苷兹2, 0 t

104. Given cot t 苷

107. 共cos t 1兲共cos t 1兲; sin2 t 108. 共sec t 1兲共sec t 1兲; tan2 t

3 兹3 ,t , find cos t. 3 2

Projects 1.

2. Consider a square as shown. Start at the point 共1, 0兲 and travel counterclockwise around the square for a distance t 共t 0兲.

VISUAL INSIGHT y B Unit circle

F

E

y

φ C

(−1, 1)

φ O

D

A(1, 0)

x

DC 苷 sin

Make use of the above circle and similar triangles to explain why a. AB is equal to tan

b. EF is equal to cot

c. OB is equal to sec

d. OF is equal to csc

(1, 1)

(1, 0) x

(−1, 0)

(−1, −1)

OD 苷 cos

(0, 1)

(0, −1)

(1, −1)

Let WSQ共t兲苷 P共x, y兲 be the point on the square determined by traveling counterclockwise a distance of t units from 共1, 0兲. For instance, WSQ共0.5兲苷共1, 0.5兲 WSQ共1.75兲苷共0.25, 1兲 Find WSQ共4.2兲 and WSQ共6.4兲. We define the square sine of t, denoted by ssin t, to be the y-value of point P. The square cosine of t, denoted by scos t, is defined to be

2.5 the x-value of point P. For example,

Find each of the following.

ssin 0.4 苷 0.4

scos 0.4 苷 1

scos 1.2 苷 0.8

scos 5.3 苷 0.7

The square tangent of t, denoted by stan t, is defined as stan t 苷

ssin t , scos t

■

a. ssin 3.2

b. scos 4.4

c. stan 5.5

d. ssin 11.2

e. scos 5.2

f. stan 6.5

scos t 苷 0

Graphs of the Sine and Cosine Functions

Section 2.5 ■

169

Graphs of the Sine and Cosine Functions

The Graph of the Sine Function The Graph of the Cosine Function

P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A10.

PS1.

Estimate, to the nearest tenth, sin

3p . [2.4] 4

PS2.

Estimate, to the nearest tenth, cos

5p . [2.4] 4

PS3. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共x兲. [1.4] PS4. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共2x兲. [1.4] PS5. Simplify:

2p 1兾3

PS6. Simplify:

2p 2兾5

The Graph of the Sine Function The trigonometric functions can be graphed on a rectangular coordinate system by plotting the points whose coordinates belong to the function. We begin with the graph of the sine function. Table 2.7 lists some ordered pairs 共x, y兲 of the graph of y 苷 sin x for 0 x 2p.

Table 2.7

Ordered Pairs of the Graph of y sin x

x

0

p 6

p 3

p 2

2p 3

5p 6

p

y sin x

0

1 2

兹3 2

1

兹3 2

1 2

0

7p 6

1 2

4p 3

3p 2

兹3 2

1

5p 3

11p 6

兹3 2

1 2

2p 0

170

Chapter 2

Trigonometric Functions In Figure 2.59, the points from the table are plotted and a smooth curve is drawn through the points. We could use decimal approximations for p on the x-axis, but it is more convenient to simply label the tick marks on the x-axis in terms of p. Note: 兹3 The y-value ⬇ 0.87. 2 y 1 7π 6 π 6

π 3

π 2

2π 3

π

5π 6

11π 6 4π 3

3π 2

5π 3

2π

x

−1

y 苷 sin x, 0 x 2

Figure 2.59

Because the domain of the sine function is the real numbers and the period is 2, the graph of y 苷 sin x is drawn by repeating the portion shown in Figure 2.59. Any part of the graph that corresponds to one period 共2兲 is one cycle of the graph of y 苷 sin x (see Figure 2.60). The Graph of y = sin x

Basic Properties

y 1 cycle 1

– 2π

π

–π

2π

3π

–1

Figure 2.60

4π

■

Domain: All real numbers

■

Range: 兵y 兩 1 y 1其

■

Period: 2p

■

Symmetry: With respect to the origin

■

x-intercepts: At multiples of p

x

The maximum value M reached by sin x is 1, and the minimum value m is 1. The amplitude of the graph of y 苷 sin x is given by Amplitude 苷 QUESTION

1 共M m兲 2

What is the amplitude of y 苷 sin x?

Recall that the graph of y 苷 a f共x兲 is obtained by stretching 共兩a兩 1兲 or shrinking 共0 兩a兩 1兲 the graph of y 苷 f共x兲. Figure 2.61 shows the graph of y 苷 3 sin x

ANSWER

Amplitude 苷

1 1 1 共M m兲苷关1 共1兲兴苷共2兲苷 1 2 2 2

2.5

171

Graphs of the Sine and Cosine Functions

that was drawn by stretching the graph of y 苷 sin x. The amplitude of y 苷 3 sin x is 3 because Amplitude 苷

1 1 共M m兲苷关3 共3兲兴苷 3 2 2

Note that for y 苷 sin x and y 苷 3 sin x, the amplitude of the graph is the coefficient of sin x. This suggests the following theorem.

y

y = 3 sin x

3

y = sin x

take note π

The amplitude is defined to be half the difference between the

2π

3π

x

−3

maximum height and the minimum height. It may not be equal to the

Figure 2.61

maximum height. For example, the graph of y 苷 4 sin x has a

Amplitude of y = a sin x

maximum height of 5 and an

The amplitude of y 苷 a sin x is 兩a兩.

amplitude of 1.

EXAMPLE 1

»

Graph y = a sin x

Graph: y 苷 2 sin x

Solution The amplitude of y 苷 2 sin x is 2. The graph of y 苷 f共x兲 is a reflection across the x-axis of y 苷 f共x兲. Thus the graph of y 苷 2 sin x is a reflection across the x-axis of y 苷 2 sin x. See the following figure.

y y = sin x 1

y π

3π

2π

x

y = 2 sin x

2

–1 π

y = sin 2x

2π

3π

x

−2 y = −2 sin x

Figure 2.62

»

Try Exercise 22, page 177

y y = sin x 1

π

2π

3π

−1 y = sin

x 2

Figure 2.63

4π x

The graphs of y 苷 sin x and y 苷 sin 2x are shown in Figure 2.62. Because one cycle of the graph of y 苷 sin 2x is completed in an interval of length , the period of y 苷 sin 2x is . x The graphs of y 苷 sin x and y 苷 sin are shown in Figure 2.63. Because 2 x one cycle of the graph of y 苷 sin is completed in an interval of length 4, the 2 x period of y 苷 sin is 4. 2

172

Chapter 2

Trigonometric Functions Generalizing the last two examples, one cycle of y 苷 sin bx, b 0, is completed as bx varies from 0 to 2. Therefore, 0 bx 2p 2 0x b 2p , is the period of y 苷 sin bx. Now we consider the b case when the coefficient of x is negative. If b 0, then using the fact that the sine function is an odd function, we have y 苷 sin共bx兲苷 sin bx, and thus the period 2 is still . This gives the following theorem. b The length of the interval,

Period of y = sin bx The period of y 苷 sin bx is

2 . 兩b兩

Table 2.8 gives the amplitude and period of several sine functions.

Table 2.8 3x 4

兩a兩

兩3兩苷 3

兩1兩苷 1

兩2兩苷 2

Period

2 兩b兩

2 苷 2

2 苷 6 1兾3

2 8 苷 3兾4 3

»

Graph: y 苷 sin x

1

Solution

Graph y = sin bx

Amplitude 苷 1

–2

–1

y 苷 2 sin

Amplitude

y

4

x 3

y 苷 3 sin共2x兲

EXAMPLE 2

2

y 苷 sin

y 苷 a sin bx

Function

x

Period 苷

2p 2p 苷苷2 兩b兩 p

•b

The graph is sketched in Figure 2.64.

»

Try Exercise 32, page 177

y 苷 sin x

Figure 2.64 Figure 2.65 shows the graph of y 苷 a sin bx for both a and b positive. Note from the graph the following properties of the function y 苷 a sin bx.

2.5 y

■

The amplitude is a.

a

■

The period is

■

For 0 x

■

The maximum value is a when x 苷

■

The minimum value is a when x 苷

■

If a 0, the graph is reflected across the x-axis.

π 2b

π b

3π 2b

2π b

x

−a

y 苷 a sin bx

Figure 2.65

2 . b

2 2 , the zeros are 0, , and . b b b

EXAMPLE 3

Graph: y 苷

y 1 2

173

Graphs of the Sine and Cosine Functions

»

p . 2b 3 . 2b

Graph y = a sin bx

1 x sin 2 3

Solution 3π

兩兩

Amplitude 苷

x

6π

1 1 苷 2 2

Period 苷

2p 苷 6p 兩1兾3兩

•b=

1 3

2p 苷 3, and 苷 6p, so the 1兾3 1兾3 1 graph has x-intercepts at (0, 0), (3p, 0), and (6p, 0). Because 0, the graph 2 1 x is the graph of y 苷 sin reflected across the x-axis, as shown in Figure 2.66. 2 3

−1 2

The zeros in the interval 0 x 6 are 0, y苷

1 x sin 2 3

Figure 2.66

»

Try Exercise 40, page 178

The Graph of the Cosine Function Table 2.9 lists some ordered pairs 共x, y兲 of the graph of y 苷 cos x for 0 x 2p.

Table 2.9

Ordered Pairs of the Graph of y cos x

x

0

p 6

p 3

p 2

y cos x

1

兹3 2

1 2

0

2p 3

1 2

5p 6

p

7p 6

兹3 兹3 1 2 2

4p 3

1 2

3p 2

5p 3

11p 6

2p

0

1 2

兹3 2

0

In Figure 2.67 on page 174, the points from the table are plotted and a smooth curve is drawn through the points.

174

Chapter 2

Trigonometric Functions y 1 2π 3 π 6

π 3

4π 3

π 2

π

5π 6

7π 6

3π 2

5π 3

11π 6

2π

x

−1

y 苷 cos x, 0 x 2

Figure 2.67

Because the domain of y 苷 cos x is the real numbers and the period is 2, the graph of y 苷 cos x is drawn by repeating the portion shown in Figure 2.67. Any part of the graph corresponding to one period 共2兲 is one cycle of y 苷 cos x (see Figure 2.68).

The Graph of y = cos x

Basic Properties

y 1 cycle

1

−2π

3π − 2

−π

π − 2

π 2

π

3π 2

2π

5π 2

3π

7π 2

4π

■

Domain: All real numbers

■

Range: 兵y兩1 y 1其

■

Period: 2p

■

Symmetry: With respect to the y-axis

■

x-intercepts: At odd multiples of

x

−1

p 2

Figure 2.68

The following two theorems concerning cosine functions can be developed using methods that are analogous to those we used to determine the amplitude and period of a sine function.

Amplitude of y = a cos x The amplitude of y 苷 a cos x is 兩a兩.

Period of y = cos bx The period of y 苷 cos bx is

2 . 兩b兩

2.5

175

Graphs of the Sine and Cosine Functions

Table 2.10 gives the amplitude and period of some cosine functions.

Table 2.10 2x 3

y 苷 3 cos

y 苷 a cos bx

y 苷 2 cos 3x

Amplitude

兩a兩

兩2兩苷 2

兩3兩苷 3

Period

2 兩b兩

2 3

2 苷 3 2兾3

2p 2p 苷苷3 兩b兩 2p兾3

•b

Function

y 1

−1

EXAMPLE 4 3 4

3 2

x

9 3 4

Graph: y 苷 cos

»

Graph y = cos bx

2 x 3

−1

Solution y 苷 cos

Amplitude 苷 1

2 x 3

Period 苷

2 3

The graph is shown in Figure 2.69.

Figure 2.69

»

Try Exercise 34, page 177

Figure 2.70 shows the graph of y 苷 a cos bx for both a and b positive. Note from the graph the following properties of the function y 苷 a cos bx.

y a 3π 2b π 2b

π b

2π b

x

■

The amplitude is a.

■

The period is

■

For 0 x

■

The maximum value is a when x 苷 0.

■

The minimum value is a when x 苷

■

If a 0, then the graph is reflected across the x-axis.

−a

y 苷 a cos bx

Figure 2.70

2 . b

2 3 , the zeros are and . b 2b 2b

EXAMPLE 5

»

Graph: y 苷 2 cos

. b

Graph a Cosine Function

x 4

Solution Amplitude 苷兩2兩苷 2

Period 苷

2p 苷8 兩p兾4兩

•b

4

Continued

䉴

176

Chapter 2

Trigonometric Functions

y

3p 苷 2 and 苷 6, so the graph 2兾4 2p兾4 has x-intercepts at (2, 0) and (6, 0). Because 2 0, the graph is the graph of The zeros in the interval 0 x 8 are

2

−2

2 4 6

8

y 苷 2 cos

10 x

x reflected across the x-axis, as shown in Figure 2.71. 4

»

−2

Try Exercise 48, page 178

y 苷 2 cos

x 4

EXAMPLE 6

Figure 2.71

»

Graph the Absolute Value of the Cosine Function

Graph y 苷兩cos x兩, where 0 x 2.

Solution

y

Because 兩cos x兩 0, the graph of y 苷兩cos x兩 is drawn by reflecting the negative portion of the graph of y 苷 cos x across the x-axis. The graph is the one shown in purple and light blue in Figure 2.72.

y = | cos x |

»

1

Try Exercise 54, page 178

π 2

π

3π 2

2π

x

In Example 7 we determine an equation for a given graph.

−1 y = cos x

Figure 2.72

EXAMPLE 7

»

Find an Equation of a Graph

The graph at the right shows one cycle of the graph of a sine or cosine function. Find an equation for the graph.

y 2

0

2

4

6

−2

Solution Because the graph obtains its maximum value at x 苷 0, start with an equation of the form y 苷 a cos bx. The graph completes one cycle in 6 units. 2p Thus the period is 6. Use the equation 6 苷 to determine b. 兩b兩 2p 兩b兩 6兩b兩苷 2p p 兩b兩苷 3 p b苷

3 6苷

• Multiply each side by 兩 b兩. • Divide each side by 6.

x

2.5

177

Graphs of the Sine and Cosine Functions

p p or as the b value. The graph has a maximum 3 3 height of 2 and a minimum height of 2. Thus the amplitude is a 苷 2. p p Substituting 2 for a and for b in y 苷 a cos bx produces y 苷 2 cos x. 3 3 We can use either

»

Try Exercise 60, page 178

Topics for Discussion 1. Is the graph of f共x兲苷兩sin x兩 the same as the graph of y 苷 sin兩x兩? Explain. 2. Explain how the graph of y 苷 cos 2x differs from the graph of y 苷 cos x. 3. Does the graph of y 苷 sin共2x兲 have the same period as the graph of y 苷 sin 2x? Explain. 4. The function h共x兲苷 a sin bt has an amplitude of 3 and a period of 4. What are the possible values of a? What are the possible values of b?

Exercise Set 2.5 In Exercises 1 to 18, state the amplitude and period of the function defined by each equation.

1 sin x 2

1. y 苷 2 sin x

2. y 苷

3. y 苷 sin 2x

4. y 苷 sin

5. y 苷

1 sin 2 x 2 x 2

7. y 苷 2 sin

9. y 苷

1 cos x 2

11. y 苷 cos

x 4

13. y 苷 2 cos

2x 3

6. y 苷 2 sin

8. y 苷

x 3

x 1 sin 2 2

10. y 苷 3 cos x

12. y 苷 cos 3x

x 3

15. y 苷 3 cos

2x 3

17. y 苷 4.7 sin 0.8pt

In Exercises 19 to 56, graph one full period of the function defined by each equation.

19. y 苷

1 sin x 2

21. y 苷 3 cos x

23. y 苷

3 cos x 2

22. y 苷

7 cos x 2

3 sin x 2

24. y 苷 3 sin x

25. y 苷 4 sin x

26. y 苷 5 cos x

27. y 苷 cos 3x

28. y 苷 sin 4x

29. y 苷 sin

31. y 苷 cos

3x 2

30. y 苷 cos x

x 2

14. y 苷

1 cos 2 x 2

33. y 苷 sin 2 x

16. y 苷

3 cos 4x 4

35. y 苷 4 cos

18. y 苷 2.3 cos 0.005pt

20. y 苷

3 x 4

34. y 苷 cos 3x

x 2

37. y 苷 2 cos

32. y 苷 sin

36. y 苷 2 cos x 3

38. y 苷

3x 4

4 cos 3x 3

178

Chapter 2

Trigonometric Functions

39. y 苷 2 sin x

41. y 苷

40. y 苷

3 x cos 2 2

43. y 苷 4 sin

x 3

42. y 苷 cos

2 x 3

63. MODEL A SOUND WAVE The following oscilloscope screen displays the signal generated by the sound of a tuning fork during a time span of 8 milliseconds.

x 1 sin 2 3

44. y 苷 3 cos

5V

3 x 2

45. y 苷 2 cos 2x

1 46. y 苷 sin 2.5x 2

47. y 苷 2 sin 1.5x

48. y 苷

兩

49. y 苷 2 sin

x 2

兩

50. y 苷

3 cos 5x 4

1 sin 3x 2

兩

51. y 苷兩2 cos 3x兩

兩

兩

−5 V

兩

兩

2x 3

兩

x 2

55. y 苷兩3 cos x兩

56. y 苷 2 cos

b. The frequency of a sound wave is defined as the reciprocal of the period of the sound wave. What is the frequency of the sound wave that produced the signal shown above? State your answer in cycles per millisecond.

兩

1 x cos 2 2

54. y 苷 3 sin

8 ms

a. Write an equation of the form V 苷 a sin bt that models the signal, where V is in volts and t is in milliseconds.

52. y 苷

兩

x 3

53. y 苷 2 sin

0

兩

64. MODEL A SOUND WAVE The following oscilloscope screen displays the signal generated by the sound of a tuning fork during a time span of 5 milliseconds.

兩

2.5 V

In Exercises 57 to 62, one cycle of the graph of a sine or cosine function is shown. Find an equation of each graph.

57.

58.

y 1

−1

0

2

π 2 π 4

y

3π 4

π

x −2

3π 2

3π

9π 2

6π

5 ms

x

−2.5 V

59.

60.

y 3

a. Write an equation of the form V 苷 a sin bt that models the signal, where V is in volts and t is in milliseconds.

y 1

9π 4 3π 4

3π x

3π 2

−1

π 2

2π 3

π

4π x 3

−3

61.

65. Sketch the graph of y 苷 2 sin 62.

y

66. Sketch the graph of y 苷 3 cos

3

1 2

1

3 2

2

2x , 3 x 6. 3

y

2

x

1 4 −3

−2

b. What is the frequency of the sound wave that produced the signal shown above? (Hint: See Exercise 63b). State your answer in cycles per millisecond.

1 2

3 4

1

x

3x , 2 x 4. 4

67. Sketch the graphs of y1 苷 2 cos

x 2

and

y2 苷 2 cos x

on the same set of axes for 2 x 4.

2.5

Graphs of the Sine and Cosine Functions

68. Sketch the graphs of y1 苷 sin 3x and

y2 苷 sin

73.

Graph y 苷 e sin x, where e ⬇ 2.71828. What is the maximum value of e sin x? What is the minimum value of e sin x? Is the function defined by y 苷 e sin x a periodic function? If so, what is the period?

74.

Graph y 苷 ecos x, where e ⬇ 2.71828. What is the maximum value of ecos x? What is the minimum value of cos x e ? Is the function defined by y 苷 ecos x a periodic function? If so, what is the period?

x 3

on the same set of axes for 2 x 4. In Exercises 69 to 72, use a graphing utility to graph each function.

69. y 苷

1 x sin x 2

70. y 苷

71. y 苷 x cos x

179

1 x sin x 2

72. y 苷 x cos x

Connecting Concepts In Exercises 75 to 77, write an equation for a sine function using the given information.

75. Amplitude 苷 2; period 苷 3

In Exercises 78 to 80, write an equation for a cosine function using the given information.

78. Amplitude 苷 3; period 苷

76. Amplitude 苷 4; period 苷 2

2

79. Amplitude 苷 3; period 苷 2.5

77. Amplitude 苷 2.5; period 苷 3.2

80. Amplitude 苷 4.2; period 苷 1

Projects CEPHEID VARIABLE STARS AND THE PERIOD-LUMINOSITY RELATIONSHIP A variable star is a star whose brightness increases and decreases in a periodic fashion. Variable stars known as Cepheid variables have very regular periods. The light curve below is for the star Delta Cephei. It shows how Delta Cephei’s brightness varies every 5.4 days. Apparent magnitude

1.

m 3.5

Delta Cephei Light Curve

4.0 5.4 days 4.5 0

2

4

6

8 Time (days)

10

12

14

t

a. During the first decade of the 1900s, the astronomer Henrietta Leavitt (1868–1921) discovered the periodluminosity relationship for Cepheid variable stars. Write a few sentences that describe this relationship. b. What is the major application of Leavitt’s discovery? (Note: The Internet is a good source of information on the period-luminosity relationship.)

180

Chapter 2

Trigonometric Functions

Graphs of the Other Trigonometric Functions

Section 2.6 ■

■

■

■

The Graph of the Tangent Function The Graph of the Cotangent Function The Graph of the Cosecant Function The Graph of the Secant Function

P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A11.

PS1.

Estimate, to the nearest tenth, tan

p . [2.2] 3

PS2.

Estimate, to the nearest tenth, cot

p . [2.2] 3

PS3. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 2f 共x兲. [1.4] PS4. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共x 2兲 3. [1.4] PS5. Simplify:

PS6. Simplify:

p 1兾2 p 3 4

兩兩

The Graph of the Tangent Function Table 2.11 lists some ordered pairs (x, y) of the graph of y 苷 tan x for 0 x

Ordered Pairs of the Graph of y tan x, 0 x

Table 2.11

p . 2

2

x

0

p 12

p 6

p 4

p 3

5p 12

p 2

y tan x

0

⬇ 0.27

兹3 ⬇ 0.58 3

1

兹3 ⬇ 1.73

⬇ 3.7

undefined

In Figure 2.73, the points from the table are plotted and a smooth curve is drawn TO REVIEW

Vertical Asymptote See page 38.

冋冊

p , y 苷 tan x increases on 2 p 关0, 兲. The y values increase slowly at first and then more rapidly as x l from 2 p the left. The line given by x 苷 is a vertical asymptote of the graph. 2 Because the tangent function is an odd function, its graph is symmetric with respect to the origin. We have used this property to produce the graph of y 苷 tan x p p for x shown in Figure 2.74. 2 2 through the points. Notice that as x increases on

0,

2.6 y

y

4

4 Vertical asymptote x=−π 2

2

π 4

π 2

x

−π

2

−π

2

π 4

4

−2 −4

181

Graphs of the Other Trigonometric Functions

π 2

x

−2 Vertical asymptote x= π

−4

2

y 苷 tan x, 0 x

p 2

y 苷 tan x,

Figure 2.73

Vertical asymptote x= π 2

p p x 2 2

Figure 2.74

Recall from Section 2.4 that the period of y 苷 tan x is p. Thus a complete graph of y 苷 tan x can be produced by replicating the graph in Figure 2.74 to the right and left, as shown in Figure 2.75.

The Graph of y tan x

Basic Properties

y

■

Domain: All real numbers except odd multiples of

■

Range: All real numbers

■

Period: p

■

Symmetry: With respect to the origin

−2

■

x-intercepts: At multiples of p

−4

■

Vertical asymptotes: At odd multiples of

■

Key points:

4 2

−π

− 3π 2

−π −π 2 4

π 4

π 2

π

3π 2

x

Figure 2.75

y = 2 tan x y

2

−

π 2

π 4

y=

1

−1

π 4

−2 −3

Figure 2.76

冊冉

冊

p p kp, 1 and kp, 1 4 4 when k is an integer.

y = tan x

3

−

冉

p 2

p 2

π 2

1 tan x 2 x

Because the tangent function is unbounded, it has no amplitude. The graph of y 苷 a tan x can be drawn by stretching 共兩a兩 1兲 or shrinking 共兩a兩 1兲 the graph of y 苷 tan x. Figure 2.76 shows the graph of three tangent functions. Because the p p point , 1 is on the graph of y 苷 tan x, we can see that the point , a must 4 4 be on the graph of y 苷 a tan x.

冉冊

冉冊

182

Chapter 2

Trigonometric Functions EXAMPLE 1

»

Graph y a tan x

Graph one period of the function y 苷

1 tan x. 3

Solution 1 tan x can be produced by shrinking the graph of y 苷 tan x 3 1 p toward the x-axis by a factor of . The point , 1 is on the graph of 3 4 p 1 1 y 苷 tan x. Thus we know that , is on the graph of y 苷 tan x. 4 3 3 The graph of y 苷

冉冊

冉冊 y 4 2

−π 2

−π

y = tan x

3

( π4 , 1) π 4

4

y = 1 tan x

−2

(

π 1 , 4 3

)

π 2

x

−4

»

Try Exercise 24, page 189

The period of y 苷 tan x is p, and the graph completes one cycle on the interval p p p x . The period of y 苷 tan bx 共b 0兲 is . The graph of y 苷 tan bx 2 2 b p p completes one cycle on the interval , . 2b 2b

冉

冊

Period of y tan bx The period of y 苷 tan bx is

QUESTION

ANSWER

p . 兩b兩

What is the period of the graph of y 苷 tan px?

p p 苷苷1 兩b兩 p

2.6

−π

2b

π 4b

π x 2b

−3a −5a

y 苷 a tan bx,

x 2b 2b

p . b

■

The period is

■

x 苷 0 is a zero.

■

The graph passes through

■

If a 0, the graph is reflected across the x-axis.

a −a

183

Figure 2.77 shows one cycle of the graph of y 苷 a tan bx for both a and b positive. Note from the graph the following properties of the function y 苷 a tan bx.

y 5a y = a tan bx 3a −π 4b

Graphs of the Other Trigonometric Functions

冉

冊冉冊

p p , a and ,a . 4b 4b

In Example 2 we graph a function of the form y 苷 a tan bx.

Figure 2.77

EXAMPLE 2

»

Graph y a tan bx x . 2

Graph one period of the function y 苷 2 tan

Solution x 1 is of the form y 苷 a tan bx, with a 苷 2 and b 苷 . 2 2 p p The period is given by 苷苷 2p. Thus one period of the graph will 兩b兩 1/2 be displayed on any interval of length 2p. In the graph below, we have chosen to sketch the function over the interval p x p. The graph p p p passes through , a 苷 , 2 苷 , 2 and 4b 4(1/2) 2 p p p ,a 苷 ,2 苷 , 2 . The function has a zero at x 苷 0, so the 4b 4(1/2) 2 graph has an x-intercept at (0, 0). The function y 苷 2 tan

冉冊冉

冉

冊冉冊冉冊

冊冉

y = 2tan x 2 y 4 2

−π

−π

π 2

2

−2

(

»

Try Exercise 32, page 190

−4 − π , −2 2

( π2 , 2)

)

π x

冊

184

Chapter 2

Trigonometric Functions

The Graph of the Cotangent Function A graph of the cotangent function y 苷 cot x is shown in Figure 2.78. Notice that its graph is similar to the graph of y 苷 tan x in that it has a period of p and is symmetric with respect to the origin. The Graph of y cot x

Basic Properties y

■

Domain: All real numbers except multiples of p

■

Range: All real numbers

■

Period: p

■

Symmetry: With respect to the origin

−2

■

x-intercepts: At odd multiples of

−4

■

Vertical asymptotes: At multiples of p

■

Key points:

4 2

π

−π

π 4

−4

2π x

π

Figure 2.78

y 4

y=−

1 cot x 2

π 2

3π 4

2 π 4 π

x

冉

p 2

冊冉

The graph of y 苷 a cot x is drawn by stretching 共兩a兩 1兲 or shrinking 共兩a兩 1兲 the graph of y 苷 cot x. The graph is reflected across the x-axis when a 0. Figure 2.79 shows the graphs of two cotangent functions. p The period of y 苷 cot x is p, and the period of y 苷 cot bx is . One cycle of 兩b兩 p the graph of y 苷 cot bx is completed on the interval 0, . b

冉冊

−2

Period of y cot bx

y = 2 cot x

−4

The period of y 苷 cot bx is

Figure 2.79 y

p . 兩b兩

Figure 2.80 shows one cycle of the graph of y 苷 a cot bx for both a and b positive. Note from the graph the following properties of the function y 苷 a cot bx.

a

−a

冊

p p kp, 1 and kp, 1 4 4 where k is an integer

π 4b

π 2b

3π 4b

y 苷 a cot bx, 0 x

Figure 2.80

π b

p b

p . b

■

The period is

■

x苷

■

The graph passes through

■

If a 0, the graph is reflected across the x-axis.

x

p is a zero. 2b

冉冊冉冊

p 3p , a and , a . 4b 4b

2.6

Graphs of the Other Trigonometric Functions

185

In Example 3 we graph a function of the form y 苷 a cot bx.

EXAMPLE 3

»

Graph y a cot bx

Graph one period of the function y 苷 2 cot

x . 3

Solution x 1 is of the form y 苷 a cot bx, with a 苷 2 and b 苷 . 3 3 p p The period is given by 苷苷 3p. Thus one period of the graph will 兩b兩 1/3 be displayed on any interval of length 3p. In the graph below, we have chosen to sketch the function over the interval 0 x 3p. The graph p p 3p passes through ,a 苷 ,2 苷 , 2 and 4b 4(1/3) 4 3p 3p 9p , a 苷 , 2 苷 , 2 . The graph has an x-intercept at 4b 4(1/3) 4 p p 3 x苷苷苷 p. 2b 2(1/3) 2 The function y 苷 2 cot

冉冊冉冊冉冊冉冊冉冊冉冊 y 4

y = 2cot x 3

2

( 3π4 , 2) 3π 4

−2

3π 2

9π 4

3π x

( 9π4 , −2)

−4

»

Try Exercise 34, page 190

The Graph of the Cosecant Function 1 , the value of csc x is the reciprocal of the value of sin x. sin x Therefore, csc x is undefined when sin x 苷 0 or when x 苷 kp, where k is an integer. The graph of y 苷 csc x has vertical asymptotes at kp. Because y 苷 csc x has period 2p, the graph will be repeated along the x-axis every 2p units. A graph of y 苷 csc x is shown in Figure 2.81 on page 186. The graph of y 苷 sin x is also shown in Figure 2.81. Note the relationships among the x-intercepts of y 苷 sin x and the asymptotes of y 苷 csc x. Because csc x 苷

186

Chapter 2

Trigonometric Functions

The Graph of y csc x

Basic Properties

y = csc x y 4 y = sin x

2 −π

3π 2

2

π 2

−π

7π 2

π

5π 2

2π

4π

3π

x

−2 −4

■

Domain: All real numbers except multiples of p

■

Range: 兵y 兩 y 1, y 1其

■

Period: 2p

■

Symmetry: With respect to the origin

■

x-intercepts: None

■

Vertical asymptotes: At multiples of p

■

Reciprocal relationship: If 共x, y兲, y 苷 0, is a point on the graph of y 苷 sin x, then (x, 1/y) is a corresponding point on the graph of y 苷 csc x.

Figure 2.81

Figure 2.82 shows one cycle of the graph of y 苷 a csc bx for both a and b positive. Note from the graph the following properties of the function y 苷 a csc bx.

y a

π 2b

π b

2π b

3π 2b

x

The period is

■

The vertical asymptotes of y 苷 a csc bx are located at the zeros of y 苷 a sin bx.

■

The graph passes through

■

If a 0, then the graph is reflected across the x-axis.

−a

y 苷 a csc bx, 0 x

2 b

Figure 2.82

take note The maximum value of a function is its largest range value. The minimum value of a function is its smallest range value.

2p . b

■

■

The vertical asymptotes of the graph of the cosecant function pass through the x-intercepts of the graph of the sine function.

■

The maximum values of the sine function are the relative minimum values of the cosecant function, and the minimum values of the sine function are the relative maximum values of the cosecant function. y

y = 2 csc 4x

4

A maximum on the sine function is a relative minimum on the cosecant function.

2

f共a兲 is greater than or equal to the values of f near x 苷 a. The value of f共b兲 is called a relative minimum of

π 8

−2

a function f, provided f共b兲 is less than or equal to the values of f near x 苷 b.

p 3p , a and , a . 2b 2b

One procedure for graphing y 苷 a csc bx is to begin by graphing y 苷 a sin bx. For instance, in Figure 2.83, we have used the graph of y 苷 2 sin 4x to produce the graph of y 苷 2 csc 4x. Observe that

The value f共a兲 is called a relative maximum of a function f, provided

冉冊冉冊

−4

y = 2 sin 4x

π 4

3π 8

π 2

5π 8

A minimum on the sine function is a relative maximum on the cosecant function.

Figure 2.83

3π 4

7π 8

π x

2.6

Graphs of the Other Trigonometric Functions

EXAMPLE 4

»

187

Graph y a csc bx

Graph one complete period of y 苷 2 csc

px . 2

Solution px and draw vertical asymptotes through the 2 x-intercepts. Use the asymptotes as guides to draw the cosecant function. px The maximum values of y 苷 2 sin are the relative minimum values of 2 px px y 苷 2 csc , and the minimum values of y 苷 2 sin are the relative 2 2 px maximum values of y 苷 2 csc . 2 Graph one period of y 苷 2 sin

y 4 y = 2 csc πx 2

2

1 −2

2

3

4 x

y = 2 sin πx 2

−4

»

Try Exercise 40, page 190

The Graph of the Secant Function 1 , the value of sec x is the reciprocal of the value of cos x. cos x p Therefore, sec x is undefined when cos x 苷 0 or when x 苷 kp, k an integer. 2 p The graph of y 苷 sec x has vertical asymptotes at kp. Because y 苷 sec x has 2 period 2p, the graph will be replicated along the x-axis every 2p units. A graph of y 苷 sec x is shown in Figure 2.84 on page 188. The graph of y 苷 cos x is also shown in Figure 2.84. Note the relationships among the x-intercepts of y 苷 cos x and the asymptotes of y 苷 sec x. Because sec x 苷

188

Chapter 2

Trigonometric Functions

The Graph of y sec x

Basic Properties

y = sec x y

■

Domain: All real numbers except odd p multiples of 2

■

Range: 兵y 兩 y 1, y 1其

■

Period: 2p

■

Symmetry: With respect to the y-axis

■

x-intercepts: None

■

Vertical asymptotes: At odd multiples of

■

Reciprocal relationship: If 共x, y兲, y 苷 0, is a point on the graph of y 苷 cos x, then (x, 1/y) is a corresponding point on the graph of y 苷 sec x.

4 2

y = cos x

−π

π

2π

3π

4π

x

−2 −4

Figure 2.84

p 2

The procedure for graphing y 苷 a sec bx is analogous to the procedure used to graph cosecant functions. First graph y 苷 a cos bx to determine its x-intercepts and its maximum and minimum points. ■

The vertical asymptotes of the graph of the secant function pass through the x-intercepts of the graph of the cosine function.

■

The maximum values of the cosine function are the relative minimum values of the secant function, and the minimum values of the cosine function are the relative maximum values of the secant function.

y a

Figure 2.85 shows one cycle of the graph of y 苷 a sec bx for both a and b positive. Note from the graph the following properties of the function y 苷 a sec bx. π 2b

π b

3π 2b

2π x b

−a

y 苷 a sec bx, 0 x

Figure 2.85

2 b

2p . b

■

The period is

■

The vertical asymptotes of y 苷 a sec bx are located at the x-intercepts of y 苷 a cos bx.

■

The graph passes through 共0, a兲,

■

If a 0, then the graph is reflected across the x-axis.

EXAMPLE 5

»

Graph: y 苷 3 sec

x 2

冉冊冉冊

p 2p , a , and ,a . b b

Graph y a sec bx

2.6 Solution

y

π −3

x and draw vertical asymptotes 2 x through the x-intercepts. Now sketch the graph of y 苷 3 sec , using the 2 asymptotes as guides for the graph, as shown in Figure 2.86. First sketch the graph of y 苷 3 cos

y = − 3 sec x 2

3

−π

189

Graphs of the Other Trigonometric Functions

2π

3π

x

4π

»

Try Exercise 44, page 190

y = −3 cos x 2

Figure 2.86

Topics for Discussion 1. Explain how you could use the graph of y 苷 sin x to produce the graph of y 苷 csc x. 2. Explain how to determine the period of y 苷 tan bx. x 3. What is the amplitude of the function y 苷 cot ? Explain. 2 4. Explain how to determine the location of the vertical asymptotes for the graph of y 苷 2 tan bx.

Exercise Set 2.6 1. For what values of x is y 苷 tan x undefined?

17. y 苷 cot px

2. For what values of x is y 苷 cot x undefined? 19. y 苷 0.5 tan

3. For what values of x is y 苷 sec x undefined? 4. For what values of x is y 苷 csc x undefined?

21. y 苷 2.4 csc

18. y 苷 cot

冉冊冉冊

px 3

p t 5

20. y 苷 1.6 cot

p t 4.25

22. y 苷 4.5 sec

冉冊冉冊 p t 2

p t 2.5

In Exercises 5 to 22, state the period of each function.

5. y 苷 sec x

6. y 苷 cot x

8. y 苷 csc x

9. y 苷 2 tan

7. y 苷 tan x x 2

10. y 苷 x 2

11. y 苷 csc 3x

12. y 苷 csc

13. y 苷 tan 3x

14. y 苷 3 cot

15. y 苷 3 sec

x 4

16. y 苷

1 cot 2x 2

In Exercises 23 to 42, sketch one full period of the graph of each function.

23. y 苷 3 tan x

25. y 苷 2x 3

1 csc 2x 2

3 cot x 2

27. y 苷 2 sec x

29. y 苷

1 csc x 2

24. y 苷

1 tan x 3

26. y 苷 4 cot x

28. y 苷

3 sec x 4

30. y 苷 2 csc x

190

Chapter 2

31. y 苷 2 tan

Trigonometric Functions

x 2

33. y 苷 3 cot

x 2

34. y 苷

x 35. y 苷 2 csc 3 37. y 苷

53.

32. y 苷 3 tan 3x

38. y 苷 3 sec

1

3π 3π 4 2

9π 4

x

3π

1

−1

x

2

−1

2x 3 55.

px 2

39. y 苷 2 sec px

40. y 苷 3 csc

41. y 苷 3 tan 2px

px 1 42. y 苷 cot 2 2

56.

y

y

1

1

2π 3

43. Graph y 苷 2 csc 3x from 2p to 2p.

4π 3

2π

8π 3

x

1

x

−1

–1

44. Graph y 苷 sec

y

1

1 cot 2x 2

3 36. y 苷 csc 3x 2

1 sec 2x 2

54.

y

x from 4p to 4p. 2

45. Graph y 苷 3 sec px from 2 to 4. In Exercises 57 to 60, graph each equation.

px 46. Graph y 苷 csc from 4 to 4. 2 47. Graph y 苷 2 cot 2x from p to p. 48. Graph y 苷

x 1 tan from 4p to 4p. 2 2

58. y 苷 sec 兩x兩

59. y 苷兩csc x兩

60. y 苷兩cot x兩

61.

49. Graph y 苷 3 tan px from 2 to 2. 50. Graph y 苷 cot

57. y 苷 tan 兩x兩

ROCKET LAUNCH An observer is 1.4 miles from the launch pad of a rocket. The rocket is launched upward as shown in the accompanying figure.

px from 4 to 4. 2

In Exercises 51 to 56, each blue graph displays one cycle of the graph of a trigonometric function. Find an equation of each blue graph.

51.

52.

y

d

h

Observer

y

x

1.4 miles

1

1

a. Write the height h, in miles, of the rocket as a function of the angle of elevation x. π 3 −1

2π 3

π

x

−π

−

π 2

π 2 −1

π

x

b. Write the distance d, in miles, from the observer to the rocket as a function of angle x. c. Graph both functions in the same viewing window with Xmin 0, Xmax p/2, Ymin 0, and Ymax 20.

2.6 d.

Graphs of the Other Trigonometric Functions

Describe the relationship between the graphs in c.

62. AVIATION A helicopter maintains an elevation of 3.5 miles and is flying toward a control tower. The angle of elevation from the tower to the helicopter is x radians.

d

a. Write the distance d, in miles, between the tower and the helicopter as a function of x. b. Find d when x 苷 1 and x 苷 1.2. Round to the nearest hundredth of a mile.

3.5 miles

x

Connecting Concepts In Exercises 63 to 70, write an equation of the form y tan bx, y cot bx, y sec bx, or y csc bx that satisfies the given conditions.

63. Tangent, period

65. Secant, period

p 3

3p 4

64. Cotangent, period

66. Cosecant, period

67. Cotangent, period 2

68. Tangent, period 0.5

69. Cosecant, period 1.5

70. Secant, period 3

p 2

5p 2

Projects 1.

2.

A TECHNOLOGY QUESTION A student’s calculator shows the display at the right. Note that the domain values 4.7123 and 4.7124 are close together, but the range values are over 100,000 units apart. Explain how this is possible. SOLUTIONS OF A TRIGONOMETRIC EQUATION How 1 many solutions does tan 苷 0 have on the interval x 1 x 1? Explain.

191

tan 4.7123 11238.43194 tan 4.7124 -90747.26955

192

Chapter 2

Section 2.7 ■

■ ■

Translations of Trigonometric Functions Addition of Ordinates Damping Factor

Trigonometric Functions

Graphing Techniques P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A12.

PS1. Find the amplitude and period of the graph of y 苷 2 sin 2x. [2.5] PS2. Find the amplitude and period of the graph of y 苷

x 2 cos . [2.5] 3 3

PS3. Find the amplitude and period of the graph of y 苷 4 sin 2px. [2.5] PS4. What is the maximum value of f 共x兲苷 2 sin x? [2.5] PS5. What is the minimum value of f 共x兲苷 3 cos 2x? [2.5] PS6. Is the graph of f 共x兲苷 cos x symmetric with respect to the origin or with respect to the y-axis? [2.5]

Translations of Trigonometric Functions

TO REVIEW

Translations of Graphs See page 58.

Recall that the graph of y 苷 f共x兲 c is a vertical translation of the graph of y 苷 f共x兲. For c 0, the graph of y 苷 f共x兲 c is the graph of y 苷 f共x兲 shifted c units down; the graph of y 苷 f共x兲 c is the graph of y 苷 f共x兲 shifted c units up. The graph in Figure 2.87 is a graph of the equation y 苷 2 sin x 3, which is a vertical translation of y 苷 2 sin x down 3 units. Note that subtracting 3 from y 苷 2 sin x changes neither its amplitude nor its period. Also, the graph of y 苷 f共x c兲 is a horizontal translation of the graph of y 苷 f共x兲. For c 0, the graph of y 苷 f共x c兲 is the graph of y 苷 f共x兲 shifted c units to the right; the graph of y 苷 f共x c兲 is the graph of y 苷 f共x兲 shifted c units to the

冉冊

left. The graph in Figure 2.88 is a graph of the equation y 苷 2 sin x

, which 4

p units to the right. Note that neither the 4 period nor the amplitude is affected. The horizontal shift of the graph of a trigonometric function is called its phase shift. is the graph of y 苷 2 sin x translated

y

y y = 2 sin πx

π unit 4

2 2

−2 2

4

x

2π –π

π 4

π

x

3π

–2 y = 2 sin x

−5

Figure 2.87

(

y = 2 sin x −

y = 2 sin πx − 3

Figure 2.88

π 4

)

2.7

193

Graphing Techniques

Because one cycle of y 苷 a sin x is completed for 0 x 2, one cycle of the graph of y 苷 a sin共bx c兲, where b 0, is completed for 0 bx c 2. Solving this inequality for x, we have 0 bx c 2 c bx c 2 c c 2 x b b b c is the phase shift for y 苷 a sin共bx c兲. The graph of the equab c tion y 苷 a sin共bx c兲 is the graph of y 苷 a sin bx shifted units horizontally. b Similar arguments apply to the remaining trigonometric functions. The number

The Graphs of y a sin(bx c) and y a cos(bx c) The graphs of y 苷 a sin共bx c兲 and y 苷 a cos共bx c兲, have Amplitude: 兩a兩

Period:

2p 兩b兩

Phase shift:

c b

To graph y 苷 a sin(bx c), shift the graph of y 苷 a sin bx horizontally

c b

units. To graph y 苷 a cos(bx c), shift the graph of y 苷 a cos bx horizontally

c b

units.

QUESTION

EXAMPLE 1 y

冉

What is the phase shift of the graph of y 苷 3 sin

»

Graph by Using a Translation

冉

y = 3 cos 2x

冊

1 x ? 2 6

Graph: y 苷 3 cos 2x

3

冊

Solution −π

π

x

−2

The phase shift is

冉

y 苷 3 cos 2x

(

y = 3 cos 2x +

Figure 2.89

π 3

)

3

冊

c 兾3 苷苷 . The graph of the equation b 2 6 is the graph of y 苷 3 cos 2x shifted units to the left, 6

as shown in Figure 2.89.

»

Try Exercise 20, page 199

ANSWER

3

194

Chapter 2

Trigonometric Functions The Graphs of y a tan(bx c) and y a cot(bx c) The graphs of y 苷 a tan共bx c兲 and y 苷 a cot共bx c兲 have Period:

p 兩b兩

Phase shift:

c b

To graph y 苷 a tan(bx c), shift the graph of y 苷 a tan bx horizontally

c b

units. To graph y 苷 a cot(bx c), shift the graph of y 苷 a cot bx horizontally

c b

units. y

y = 2 cot (3x − 2)

EXAMPLE 2

4

»

Graph by Using a Translation

Graph one period of the function y 苷 2 cot共3x 2兲.

2

π 6

−2

π 2

π 3

x

Solution The phase shift is

−4 y = 2 cot (3x)

c 2 2 苷苷 b 3 3

• 3x 2 3x (2)

The graph of y 苷 2 cot共3x 2兲 is the graph of y 苷 2 cot共3x兲 shifted 2 unit to the right, as shown in Figure 2.90. 3

Figure 2.90

»

Try Exercise 22, page 199

In Example 3 we use both a horizontal translation and a vertical translation to graph a function of the form y 苷 a sin(bx c) d. EXAMPLE 3

Graph: y 苷

1 y = sin x 2

y

π

2π

−1

π

( 4 , −2) −3

y=

π 1 sin x − − 2 4 2

(

Figure 2.91

Graph y a sin(bx c) d

冉冊

1 sin x 2 4

2

Solution

3π 2 π 2

»

)

x

c 兾4 苷苷 . The vertical shift is 2 units down. b 1 4 1 1 The graph of y 苷 sin x 2 is the graph of y 苷 sin x shifted 2 4 2 4 units to the right and 2 units down, as shown in Figure 2.91. The phase shift is

冉冊

»

Try Exercise 40, page 199

In Example 4 we use both a horizontal translation and a vertical translation to graph a function of the form y 苷 a cos(bx c) d.

2.7 EXAMPLE 4 y

(− 12 , 1)

(

y = − 2 cos πx +

» 冉

π +1 2

)

Graph y a cos(bx c) d

Graph: y 苷 2 cos x

3

195

Graphing Techniques

2

冊

1

Solution

2

c 兾2 1 苷苷 . The vertical shift is 1 unit up. The b 2 graph of y 苷 2 cos x 1 is the graph of y 苷 2 cos x shifted 2 1 unit to the left and 1 unit up, as shown in Figure 2.92. 2 The phase shift is

1

2

–2

3

y = − 2 cos π x

Figure 2.92

x

冉

冊

»

Try Exercise 42, page 199

The following application involves a function of the form y 苷 cos共bx c兲 d.

EXAMPLE 5

»

A Mathematical Model of a Patient’s Blood Pressure

冉

冊

10 t 112, 0 t 20, gives the 3 3 blood pressure, in millimeters of mercury (mm Hg), of a patient during a 20-second interval. The function bp共t兲苷 32 cos

a.

Find the phase shift and the period of bp.

b.

Graph one period of bp.

c.

What are the patient’s maximum (systolic) and minimum (diastolic) blood pressure readings during the given time interval?

d.

What is the patient’s pulse rate in beats per minute?

Solution

冉冊冉冊

p c 3 Phase shift 苷苷 b 10p 3

a.

Period 苷

b.

苷 0.1

2p 2p 苷苷 0.6 second 兩b兩 10p 3

冉冊

冉冊

The graph of bp is the graph of y1 苷 32 cos

冉

10 t shifted one tenth of a 3

10 t 3 3 top of the next page, and 112 units upward. unit to the right, shown by y2 苷 32 cos

冊

in the graph at the Continued 䉴

Chapter 2

Trigonometric Functions

Blood pressure in mm Hg

196

y1

−0.1

160

bp

120 112

80 40

y2

0.1

−20

0.6

t

Time in seconds

冉

冊

10 t has a maximum of 32 and a 3 3 minimum of 32. Thus the patient’s maximum blood pressure is 32 112 苷 144 mm Hg, and the patient’s minimum blood pressure is 32 112 苷 80 mm Hg. The function y2 苷 32 cos

c.

From a. we know that the patient has 1 heartbeat every 0.6 second. Therefore, during the given time interval, the patient has a pulse rate of

d.

冉

冊冉

1 heartbeat 0.6 second

冊

60 seconds 苷 100 heartbeats per minute 1 minute

»

Try Exercise 64, page 200

The translation techniques used to graph sine and cosine functions can also be used to graph secant and cosecant functions.

EXAMPLE 6 y = 2 csc (2x − π)

»

Graph a Cosecant Function

Graph: y 苷 2 csc共2x 兲

y

Solution

4

c p p 苷苷 . The graph of the equation b 2 2 p y 苷 2 csc共2x 兲 is the graph of y 苷 2 csc 2x shifted units to the 2 p right. Sketch the graph of the equation y 苷 2 sin 2x shifted units to the 2 right. Use this graph to draw the graph of y 苷 2 csc共2x p兲, as shown in The phase shift is

−π 2

−2

π 2

π

3π 2π 2

x

y = 2 sin (2x − π)

−4

Figure 2.93.

Figure 2.93

»

Try Exercise 48, page 199

2.7

197

Graphing Techniques

Addition of Ordinates Given two functions g and h, the sum of the functions is the function f defined by f共x兲苷 g共x兲 h共x兲. The graph of the sum f can be obtained by graphing g and h separately and then geometrically adding the y-coordinates of each function for a given value of x. It is convenient, when we are drawing the graph of the sum of two functions, to pick zeros of the functions. EXAMPLE 7

»

Graph the Sum of Two Functions

Graph: y 苷 x cos x

Solution Graph g共x兲苷 x and h共x兲苷 cos x on the same coordinate grid. Then add the y-coordinates geometrically point by point. The figure below shows the results of adding, by using a ruler, the y-coordinates of the two y y = g(x) + h(x) = x + cos x functions for selected values of x. 5π Notice that when h共x兲苷 cos x 苷 0, 2 the graph of y intersects the graph of g共x兲苷 x. 2π d3 3π 2

g(x) = x

π π 2 1

−1

d2

h(x) = cos x d1

π 2

d3 d2 π

3π 2

2π

5π x 2

»

Try Exercise 52, page 199

y

In Example 8 we graph y 苷 sin x cos x by geometrically adding the y values of selected points on the graphs of g(x) 苷 sin x and h(x) 苷 cos x.

y = sin x − cos x h(x) = − cos x

1

EXAMPLE 8

»

Graph the Difference of Two Functions

Graph y 苷 sin x cos x for 0 x 2. π 2 −1

π

3π 2

g(x) = sin x

Figure 2.94

2π x

Solution Graph g共x兲苷 sin x and h共x兲苷 cos x on the same coordinate grid. For selected values of x, add g共x兲 and h共x兲 geometrically. Now draw a smooth curve through the points. See Figure 2.94.

»

Try Exercise 56, page 199

198

Chapter 2

Trigonometric Functions

Damping Factor 1 1 x in f共x兲苷 x cos x is referred to as the damping factor. In the next 4 4 example we analyze the role of the damping factor.

The factor f(x) =

6 y=

1 x cos x 4

1 x 4

EXAMPLE 9

0

8π

y=− −6

»

Graph f共x兲苷

1 x 4

Graph the Product of Two Functions

1 x cos x, x 0, and analyze the role of the damping 4

factor.

Figure 2.95

Solution Figure 2.95 shows that the graph of f intersects

take note Replacing the damping factor can

1 x for x 苷 0, 2, 4, . . . 4

■

the graph of y 苷

■

the graph of y 苷

■

the x-axis for x 苷

make a dramatic change in the graph of a function. For instance, the graph of f 共x兲苷 20.1x cos x approaches 0 as x approaches . 1 y=2

−0.1x

0

1 x for x 苷 , 3, 5, . . . 4

• Because cos x 1 for x (2n 1)

1 3 5 , , , . . . 2 2 2

• Because cos x 0 for x

8π

2n 1

2

Figure 2.95 also shows that the graph of f lies on or between the lines

y = −2 −0.1x −1

• Because cos x 1 for x 2n

y苷

f (x) = 2 −0.1x cos x

1 x 4

and

y苷

1 x 4

• Because 兩cos x兩 1 for all x

»

Try Exercise 78, page 202

Topics for Discussion 1. The maximum value of f共x兲苷 3 sin x 4 is 7. Thus f has an amplitude of 7. Do you agree? Explain. 2. The graph of y 苷 sec x has a period of 2. What is the period of the graph of y 苷兩sec x兩? 3. The zeros of y 苷 sin x are the same as the zeros of y 苷 x sin x. Do you agree? Explain.

冉

4. What is the phase shift of the graph of y 苷 tan 3x

冊

? 6

2.7

199

Graphing Techniques

Exercise Set 2.7 In Exercises 1 to 8, find the amplitude, phase shift, and period for the graph of each function.

冉冊冉冊冉冊

1. y 苷 2 sin x

2

2. y 苷 3 sin共x 兲

3. y 苷 cos 2x

4

4. y 苷

2x 3 6

6. y 苷

5. y 苷 4 sin

7. y 苷

5 cos共3x 2兲 4

冉冊冉冊冉冊

3 x cos 4 2 3

3 x 3 sin 2 4 4

8. y 苷 6 cos

x 3 6

In Exercises 9 to 16, find the phase shift and the period for the graph of each function.

冉

冊冉冊冉冊冉冊

9. y 苷 2 tan 2x

4

x 3

11. y 苷 3 csc

8

13. y 苷 2 sec 2x

x 3 4

15. y 苷 3 cot

10. y 苷

冉冊冉冊冉冊冉冊

1 x tan 2 2

12. y 苷 4 csc 3x

14. y 苷 3 sec

16. y 苷

6

x 4 2

3 cot 2x 2 4

In Exercises 17 to 32, graph one full period of each function.

冉冉冉冉冉冉

17. y 苷 sin x 2 19. y 苷 cos

x 2 3

21. y 苷 tan x 4 23. y 苷 2 cot

x 2 8

25. y 苷 sec x

27. y 苷 csc

冊冊冊冊冊冊

4

x 3 2

冉冊冉冊

18. y 苷 sin x 6 20. y 苷 cos 2x

冉冉

29. y 苷 2 sin

x 2 3 3

31. y 苷 3 cos 3x

30. y 苷

冉冉

冊冊

3 sin 2x 2 4

32. y 苷 4 cos

3x 2 2

In Exercises 33 to 50, graph each function by using translations.

33. y 苷 sin x 1

34. y 苷 sin x 1

35. y 苷 cos x 2

36. y 苷 2 sin x 3

37. y 苷 sin 2x 2

38. y 苷 cos

x 2 2

39. y 苷 4 cos共 x 2兲 1

冉

40. y 苷 2 sin

冊

x 1 2 2

41. y 苷 sin共 x 1兲 2 42. y 苷 3 cos共2 x 3兲 1

冉冊冉冊

43. y 苷 sin x

2

44. y 苷 2 cos x

45. y 苷 tan

x 4 2

47. y 苷 sec 2x 2

3

4

冊冊

49. y 苷 csc

x 1 2

3

1 2 3

46. y 苷 cot 2x 3

48. y 苷 csc

x 4 3

冉冊

50. y 苷 sec x

2

1

22. y 苷 tan共x 兲

24. y 苷

冉

3 cot 3x 2 4

冊

In Exercises 51 to 56, graph the given function by using the addition-of-ordinates method.

51. y 苷 x sin x

52. y 苷

x cos x 2

53. y 苷 x sin 2x

54. y 苷

2x sin x 3

55. y 苷 sin x cos x

56. y 苷 cos x sin x

26. y 苷 csc共2x 兲

冉

28. y 苷 sec 2x

6

冊

200

Chapter 2

Trigonometric Functions

In Exercises 57 to 62, each blue graph displays one cycle of the graph of a trigonometric function. Find an equation of each blue graph.

57.

62.

y

1

y

1

−

π 2

π 2

π

3π 2

x

−1 π 6

2π 3

x

7π 6

−1

58.

63. RETAIL SALES The manager of a major department store finds that the number of men’s suits S, in hundreds, that the store sells is given by

y

冉

2

S 苷 4.1 cos

1 π

59.

3π 2

2π

3π x

5π 2

y

冊

t 1.25 7 6

where t is time measured in months, with t 苷 0 representing January 1. a. Find the phase shift and the period of S.

2

b. Graph one period of S. 1

π

2π

3π 4π

5π

6π

c. Use the graph from b. to determine in which month the store sells the most suits.

x

−1

64. RETAIL SALES The owner of a shoe store finds that the number of pairs of shoes S, in hundreds, that the store sells can be modeled by the function

−2

冉

S 苷 2.7 cos 60.

y

2

冊

7 t 4 6 12

where t is time measured in months, with t 苷 0 representing January 1.

1

a. Find the phase shift and the period of S. −1

π 4

π 2

3π 4

π

x

b. Graph one period of S. c. Use the graph from b. to determine in which month the store sells the most shoes.

−2

61.

y

3

3π 2 π 2 −3

π

2π

5π 2

x

65. CARBON DIOXIDE LEVELS Because of seasonal changes in vegetation, carbon dioxide 共CO2兲 levels, as a product of photosynthesis, rise and fall during the year. Besides the naturally occurring CO2 from plants, additional CO2 is given off as a pollutant. A reasonable model of CO2 levels in a city for the years 1990–2006 is given by y 苷 2.3 sin 2 t 1.25t 315, where t is the number of years since 1990 and y is the concentration of CO2 in parts per million (ppm). Find the difference in CO2 levels between the beginning of 1990 and the beginning of 2006.

2.7 66. ROTATIONAL MOTION TO LINEAR MOTION A Scotch Yoke is a mechanical device that is used to convert rotational motion into linear motion. In the accompanying figure, the starter of an engine rotates point A in a counterclockwise direction at a constant rate. y

201

Graphing Techniques

lag the voltage by 0.005 second. Write an equation for the voltage and an equation for the current. V

Voltage 1/60 s

i

Current

5

180

A

−1

1

Unit circle

−0.005

Piston B

t 2

3

4

5

6

0.01

t

1 60

x −5

−180

Yoke

t

OA 苷 1; OB 苷 6 when t 苷 0

As A rotates, it also moves up and down in the yoke, which causes the piston to move to the left and to the right. a. Write the coordinates of A in terms of angle t. b. Point B is on the right end of the piston. Write the coordinates of B in terms of t.

400 m

Describe the motion of B as A makes one complete revolution.

67. HEIGHT OF A PADDLE The paddle wheel on a riverboat is shown in the accompanying figure. Write an equation for the height of a paddle relative to the water at time t. The radius of the paddle wheel is 7 feet, and the distance from the center of the paddle wheel to the water is 5 feet. Assume that the paddle wheel rotates at 5 revolutions per minute and that the paddle is at its highest point at t 苷 0. Graph the equation for 0 t 0.20 minute.

7 ft

s

5 ft

68. VOLTAGE AND AMPERAGE The graphs of the voltage and amperage of an alternating household circuit are shown in the following figures, where t is measured in seconds. Note that there is a phase shift between the graph of the voltage and the graph of the current. The current is said to

400 m

θ

s

Lighthouse Wall

Side view

Top view

70. HOURS OF DAYLIGHT The duration of daylight for a region is dependent not only on the time of year but also on the latitude of the region. The following graph gives the daylight hours for a one-year period at various latitudes. Assuming that a sine function can model these curves, write an equation for each curve.

Duration of daylight, in hours

c.

69. A LIGHTHOUSE BEACON The beacon of a lighthouse 400 meters from a straight sea wall rotates at 6 revolutions per minute. Using the accompanying figures, write an equation expressing the distance s, measured in meters, in terms of time t. Assume that when t 苷 0, the beam is perpendicular to the sea wall. Sketch a graph of the equation for 0 t 10 seconds.

D 15 14 13 12

10° N 20° N 30° N 40° N

11 10 9 8 7 J

F M A M

J

J

A

S

O N D

t

202

Chapter 2

Trigonometric Functions

71. TIDES During a 24-hour day, the tides raise and lower the depth of water at a pier as shown in the figure below. Write an equation of the form f 共t兲苷A cos Bt d that gives the depth of the water at time t, and find the depth of the water at 6 P.M.

Depth, in feet

f(t)

In Exercises 73 to 80, use a graphing utility to graph each function.

73. y 苷 sin x cos

x 2

75. y 苷 2 cos x sin

12 9

77. y 苷

6

79. y 苷 x sin 12

18

24

x 2

76. y 苷

x sin x 2

3 6

74. y 苷 2 sin 2x cos x

t

x 1 cos 2x sin 2 2

78. y 苷 x cos x

x 2

80. y 苷

x x cos 2 2

t is the number of hours from 6 A.M.

BEATS When two sound waves have approximately the 72. TEMPERATURE During a summer day, the ground temperature at a desert location was recorded and graphed as a function of time, as shown in the following figure. The graph can be approximated by f 共t兲苷 A cos 共bt c兲 d. Find the equation and approximate the temperature (to the nearest degree) at 1:00 P.M. f(t) 120

same frequency, the sound waves interfere with one another and produce phenomena called beats, which are heard as variations in the loudness of the sound. A piano tuner can use these phenomena to tune a piano. By striking a tuning fork and then tapping the corresponding key on a piano, the piano tuner listens for beats and adjusts the tension in the string until the beats disappear. Use a graphing utility to graph the functions in Exercises 81 and 82, which are based on beats.

冉冊冉冊

81. y 苷 sin共5 x兲 sin

80 40

x 2

82. y 苷 sin共17 x兲 sin −15

−9

−3

3

9

15

21 t

x 2

t is the number of hours from 6 A.M.

Connecting Concepts 83. Find an equation of the sine function with amplitude 2, period , and phase shift . 3 84. Find an equation of the cosine function with amplitude 3, period 3, and phase shift . 4 85. Find an equation of the tangent function with period 2 and phase shift . 2 86. Find an equation of the cotangent function with period and phase shift . 2 4

87. If g共x兲苷 x 2 2 and h共x兲苷 cos x, find g 关h共x兲兴. 88. If g共x兲苷 sin x and h共x兲苷 x 2 2x 1, find h关 g共x兲兴. In Exercises 89 to 92, use a graphing utility to graph each function.

89. y 苷

sin x x

91. y 苷兩x兩 sin x

90. y 苷 2 sec 92. y 苷兩x兩 cos x

x 2

2.8

203

Harmonic Motion — An Application of the Sine and Cosine Functions

Projects 1.

Write an equation that could be used to model the rabbit population shown by the graph below, where t is measured in months.

PREDATOR-PREY RELATIONSHIP Predator-prey interactions can produce cyclic population growth for both the predator population and the prey population. Consider an animal reserve where the rabbit population r is given by

r

冉冊

r共t兲苷 850 210 sin

and the wolf population w is given by

冉

w共t兲苷 120 30 sin

1000

t 6

800 600

冊

400

t2 6

200

where t is the number of months after March 1, 2006. Graph r共t兲 and w共t兲 on the same coordinate system for 0 t 24. Write a few sentences that explain a possible relationship between the two populations.

Section 2.8 ■ ■

Simple Harmonic Motion Damped Harmonic Motion

3

6

9

12

15

18

21

t

Harmonic Motion—An Application of the Sine and Cosine Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A13.

PS1. Find the reciprocal of

2p . 3

PS2. Find the reciprocal of

2 . 5

PS3. Evaluate a cos 2 t for a 苷 4 and t 苷 0. [2.4] PS4. Evaluate

冑

k for k 苷 18 and m 苷 2. m

冉冑冊

PS5. Evaluate 4 cos

k t for k 苷 16, m 苷 4, and t 苷 2. [2.4] m

PS6. Write an equation of the form y 苷 a cos bx whose graph has an amplitude of 4 and a period of 2. [2.5]

The vibratory motion of a mass on a spring is called harmonic motion. In this section, we will examine two types of harmonic motion.

204

Chapter 2

Trigonometric Functions

Simple Harmonic Motion

Equilibrium position

Figure 2.96

We will consider a mass on a spring to illustrate vibratory motion. Assume that we have placed a mass on a spring and allowed the spring to come to rest, as shown in Figure 2.96. The system is said to be in equilibrium when the mass is at rest. The point of rest is called the origin of the system. We consider the distance above the equilibrium point as positive and the distance below the equilibrium point as negative. If the mass is now lifted a distance a and released, the mass will oscillate up and down in periodic motion. If there is no friction, the motion repeats itself in a certain period of time. The number of times the mass oscillates in 1 unit of time is called the frequency f of the motion, and the time one oscillation takes is the period p of the motion. The motion is referred to as simple harmonic motion. Figure 2.97 shows the displacement y of the mass for one oscillation at times p p 3p t 苷 0, , , , and p. 4 2 4

take note The graph of the displacement y as a function of the time t is a cosine curve. y

Equilibrium

a t

–a

Figure 2.97

The frequency and the period are related by the formulas f苷

1 p

and

p苷

1 f

The maximum displacement from the equilibrium position is called the amplitude of the motion. Vibratory motion can be quite complicated. However, the simple harmonic motion of a mass on a spring can be described by one of the following equations. Simple Harmonic Motion

take note Function (1) is used if the displacement from the origin is at a maximum at time t 苷 0. Function (2) is used if the displacement at time t 苷 0 is zero.

Simple harmonic motion can be modeled by one of the following functions:

y 苷 a cos 2 ft (1)

or

y 苷 a sin 2 ft

(2)

where 兩a兩 is the amplitude (maximum displacement), f is the frequency, the period, y is the displacement, and t is the time.

1 is f

2.8

Harmonic Motion — An Application of the Sine and Cosine Functions QUESTION

205

A simple harmonic motion has a frequency of 2 cycles per second. What is the period of the simple harmonic motion?

EXAMPLE 1

»

Find the Equation of Motion of a Mass on a Spring

A mass on a spring has been displaced 4 centimeters above the equilibrium 1 point and released. The mass is vibrating with a frequency of cycle per 2 second. Write the equation of simple harmonic motion, and graph three cycles of the displacement as a function of time.

y

4

2

4

6

t

Solution Because the maximum displacement is 4 centimeters when t 苷 0, use y 苷 a cos 2 ft.

−4

y 苷 a cos 2 ft y 苷 4 cos t

苷 4 cos 2p

Figure 2.98

• Equation for simple harmonic motion

冉冊

1 t 2

• a 4, f

1 2

苷 4 cos pt The equation of simple harmonic motion is y 苷 4 cos pt. See Figure 2.98.

»

Try Exercise 20, page 208

From physical laws determined by experiment, the frequency of oscillation of a mass on a spring is given by f苷

1 2p

冑

k m

where k is a spring constant determined by experiment and m is the mass. The simple harmonic motion of the mass on the spring (with maximum displacement at t 苷 0) can then be described by

冉冑冊

y 苷 a cos 2pft 苷 a cos 2p y 苷 a cos

冑

k t m

1 2p

k t m (3)

The equation of the simple harmonic motion for zero displacement at t 苷 0 is y 苷 a sin

ANSWER

p苷

冑

k t m

1 1 苷 second per cycle f 2

(4)

206

Chapter 2

Trigonometric Functions

EXAMPLE 2

»

Find the Equation of Motion of a Mass on a Spring

A mass of 2 units is in equilibrium suspended from a spring with a spring constant of k 苷 18. The mass is pulled down 0.5 unit and released. Find the period, frequency, and amplitude of the resulting simple harmonic motion. Write the equation of the motion, and graph two cycles of the displacement as a function of time.

Solution At the start of the motion, the displacement is at a maximum but in the negative direction. The resulting motion is described by Equation (3), using a 苷 0.5, k 苷 18, and m 苷 2.

y

1

y 苷 a cos

0.5

− 0.5

π

π 3

y 苷 0.5 cos 3t

k t 苷 0.5 cos m

冑

18 t 2

y 苷 0.5 cos 3t

t

Period:

−1

冑

• Equation of motion

2p 2p 苷兩b兩 3

Frequency:

Figure 2.99

• Substitute for a, k, and m.

1 3 苷 period 2p

Amplitude: 兩a兩苷兩0.5兩苷 0.5 See Figure 2.99.

»

Try Exercise 28, page 208

Damped Harmonic Motion The previous examples have assumed that there is no friction within the spring and no air resistance. If we consider friction and air resistance, then the motion of the mass tends to decrease as t increases. The motion is called damped harmonic motion and can be modeled by an equation in the form f(t) 苷 aektcos wt. EXAMPLE 3

»

Model Damped Harmonic Motion

A mass on a spring has been displaced 14 inches below the equilibrium point and released. The damped harmonic motion of the mass is given by f共t兲苷 14e0.4t cos 2t, t 0 where f共t兲 is measured in inches and t is the time in seconds. a.

Find the values of t for which f共t兲苷 0.

b.

Use a graphing utility to determine how long it will be until the absolute value of the displacement of the mass is always less than 0.01 inch.

2.8

take note The motion in Example 3 is not

Solution f共t兲苷 0 if and only if cos 2t 苷 0, and cos 2t 苷 0 when 2t 苷

a.

periodic in a strict sense because but tends to diminish as t increases. However, because the motion cycles every p seconds, we call p the pseudoperiod of the motion. In general, the damped

We need to find the smallest value of t for which the absolute value of the displacement of the mass is always less than 0.01. Figure 2.101 shows a graph of f using a viewing window of 0 t 20 and 0.01 f共t兲 0.01. Use the TRACE or INTERSECT feature to determine that t ⬇ 17.59 seconds.

b.

harmonic motion modeled by f 共t兲苷 aekt cos t has

frequency f 苷

冉冊

p np. 2

p p n ⬇ 0.79 n共1.57兲. The 4 2 displacement f共t兲苷 0 first occurs when t ⬇ 0.79 second. The graph of f in Figure 2.100 confirms these results. Figure 2.100 also shows that the graph of f lies on or between the graphs of y 苷 14e0.4t and y 苷 14e0.4t. Recall from Section 2.7 that 14e0.4t is the damping factor. Therefore, f共t兲苷 0 when t 苷

the motion does not repeat exactly

pseudoperiod p 苷

207

Harmonic Motion — An Application of the Sine and Cosine Functions

2 and

. 2

0.01

10 y = −14 e−0.4t cos 2t y = 14e−0.4t 3π

0

20

0

y = −14 e−0.4t − 10

t = 17.59

−0.01

f共t兲苷 14e0.4t cos 2t

Figure 2.100

Figure 2.101

»

Try Exercise 34, page 209

The examples in this section involve the harmonic motion of a mass on a spring; however, there are other types of applications that involve harmonic motion. Some of these applications are illustrated in Exercises 29–32 on pages 208 and 209.

Topics for Discussion 1. The period of a simple harmonic motion is the same as the frequency of the motion. Do you agree? Explain. 2. In the simple harmonic motion modeled by y 苷 3 cos 2 t, does the displacement y approach 0 as t increases without bound? Explain. 3. Explain how you know whether to use y 苷 a cos 2 ft

or

to model a particular harmonic motion.

y 苷 a sin 2pft

208

Chapter 2

Trigonometric Functions

Exercise Set 2.8 In Exercises 1 to 8, find the amplitude, period, and frequency of the simple harmonic motion.

1. y 苷 2 sin 2t

3. y 苷 3 cos

2. y 苷

2t 3

4. y 苷 4 sin 3t

5. y 苷 4 cos t

7. y 苷

t 2 cos 3 3

6. y 苷 2 sin

t 3

20. Amplitude 3 inches, frequency

21. Amplitude 2.5 inches, frequency 0.5 cycle per second 22. Amplitude 5 inches, frequency

23. Amplitude

1 cycle per second 8

1 inch, period 3 seconds 2

24. Amplitude 5 centimeters, period 5 seconds

3 t sin 4 2

8. y 苷 5 cos 2 t 25. Amplitude 4 inches, period

In Exercises 9 to 12, write an equation for the simple harmonic motion that satisfies the given conditions. Assume that the maximum displacement occurs at t 0. Sketch a graph of the equation.

9. Frequency 苷 1.5 cycles per second, a 苷 4 inches 10. Frequency 苷 0.8 cycle per second, a 苷 4 centimeters 11. Period 苷 1.5 seconds, a 苷

1 cycle per second

3 feet 2

12. Period 苷 0.6 second, a 苷 1 meter In Exercises 13 to 18, write an equation for the simple harmonic motion that satisfies the given conditions. Assume zero displacement at t 0. Sketch a graph of the equation.

13. Amplitude 2 centimeters, period p seconds 14. Amplitude 4 inches, period

seconds 2

seconds 2

26. Amplitude 2 centimeters, period seconds 27. SIMPLE HARMONIC MOTION A mass of 32 units is in equilibrium suspended from a spring. The mass is pulled down 2 feet and released. Find the period, frequency, and amplitude of the resulting simple harmonic motion. Write an equation of the motion. Assume a spring constant of k 苷 8. 28. SIMPLE HARMONIC MOTION A mass of 27 units is in equilibrium suspended from a spring. The mass is pulled down 1.5 feet and released. Find the period, frequency, and amplitude of the resulting simple harmonic motion. Write an equation of the motion. Assume a spring constant of k 苷 3. 29. SOUND MODEL A trumpet player plays a musical note and sustains the volume of the sound for several seconds. A model of the sound wave is V(t) 苷 0.1 sin 392t, where V(t) is the variation in air pressure (from normal air pressure) after t seconds, measured in pounds per square inch.

15. Amplitude 1 inch, period 2 seconds 16. Amplitude 3 centimeters, period 1 second 17. Amplitude 2 centimeters, frequency 1 second

a. Find the frequency and period of the sound wave. Assume the frequency is in cycles per second. b. If the trumpet player increases the volume of the sound, how does the model need to change?

18. Amplitude 4 inches, frequency 4 seconds In Exercises 19 to 26, write an equation for the simple harmonic motion that satisfies the given conditions. Assume that the maximum displacement occurs when t 0.

19. Amplitude

1 2 centimeter, frequency cycle per second 2

30. TIDAL CYCLE During a 12.5-hour period, the water level at Grays Harbor, Washington, started at mean sea level, rose to 9.3 feet above sea level, dropped to 9.3 feet below sea level, and then returned to mean sea level. Find a simple harmonic motion equation that models the height h of the tide above or below mean sea level for this 12.5-hour period.

2.8

Harmonic Motion — An Application of the Sine and Cosine Functions

31. AMUSEMENT PARK RIDE A person riding on a circular Ferris wheel reaches a maximum height of 78 feet and a minimum height of 4 feet above ground level. When in uniform motion, the Ferris wheel makes one complete revolution every 45 seconds. Find an equation that gives the height h above ground level of a person riding the Ferris wheel as a function of the time t. Assume the Ferris wheel is in motion and the person is 4 feet above ground level at t 苷 0 seconds. 32. TIME IN MOTION A clock with a minute hand that is 12 inches long is hanging on a wall. At precisely 8:00 A.M. the distance from the tip of the minute hand to the floor is 6.5 feet. Thirty minutes later, this distance is 4.5 feet. Find an equation that gives the distance d, in feet, from the floor to the tip of the minute hand as a function of the number of minutes t after 8:00 A.M.

209

In Exercises 33 to 40, each of the equations models the damped harmonic motion of a mass on a spring. a. Find the number of complete oscillations that occur during the time interval 0 t 10 seconds. b. Use a graph to determine how long it will be (to the nearest tenth of a second) until the absolute value of the displacement of the mass is always less than 0.01.

33. f 共t兲苷 4e0.1t cos 2t

34. f 共t兲苷 12e0.6t cos t

35. f 共t兲苷 6e0.09t cos 2 t

36. f 共t兲苷 11e0.4t cos t

37. f 共t兲苷 e0.5t cos 2 t

38. f 共t兲苷 e0.2t cos 3 t

39. f 共t兲苷 e0.75t cos 2 t

40. f 共t兲苷 et cos 2 t

Connecting Concepts 41. Assume that a mass of m pounds on the end of a spring is oscillating in simple harmonic motion. What will be the effect on the period of the motion if the mass is increased to 9m? 42. A mass on the end of a spring oscillates in simple harmonic motion. What will be the effect on the frequency of the motion if the spring is replaced with a stiffer spring that has a spring constant four times that of the original spring?

In Exercises 43 to 46, use a graphing utility to determine whether both of the given functions model the same damped harmonic motion.

冉冊

4 g共t兲苷 e0.2t共cos t sin t兲

43. f 共t兲苷兹2 e0.2t sin t

44. f 共t兲苷 5e0.3t cos共t 0.927295兲 g共t兲苷 e0.2t共3 cos t 4 sin t兲 45. f 共t兲苷 13e0.4t cos共t 1.176005兲 g共t兲苷 e0.4t 共5 cos t 12 sin t兲 46. f 共t兲苷兹2 e0.2t cos 共t 0.785398兲 g共t兲苷 e0.2t共cos t sin t兲

Projects 1.

THREE TYPES OF DAMPED HARMONIC MOTION In some cases the damped harmonic motion of a mass on the end of a spring does not cycle about the equilibrium point. The following three functions illustrate different types of damped harmonic motion. Graph each function and then write a few sentences that explain the major differences among the motions. f 共t兲苷共0.5t 1兲e0.5t

g共t兲苷 2e0.4t 5et 0.2t

h共t兲苷 4e 2.

cos 2 t

LOGARITHMIC DECREMENT If a damped harmonic motion is modeled by f 共t兲苷 aet cos t

then the ratio of any two consecutive relative maxima of the motion is a constant, . a. Determine for the damped harmonic motion modeled by f 共t兲苷 14e0.4t cos 2t,

t0

2 is called the logarithmic decrement of the motion. Compute for the damped harmonic motion in the equation in a. How does ln g compare with ?

b. The constant 苷

210

Chapter 2

Trigonometric Functions

Exploring Concepts with Technology Sinusoidal Families Some graphing calculators have a feature that allows you to graph a family of functions easily. For instance, entering Y1={2,4,6}sin(X) in the Y= menu and pressing the GRAPH key on a TI-83/TI-83 Plus/TI-84 Plus calculator produces a graph of the three functions y 苷 2 sin x, y 苷 4 sin x, and y 苷 6 sin x, all displayed in the same window. 1. Use a graphing calculator to graph Y1={2,4,6}sin(X). Write a sentence that indicates the similarities and differences among the three graphs. 2. Use a graphing calculator to graph Y1=sin({,2,4}X). Write a sentence that indicates the similarities and differences among the three graphs. 3. Use a graphing calculator to graph Y1=sin(X+{/4,/6,/12}). Write a sentence that indicates the similarities and differences among the three graphs. 4. A student has used a graphing calculator to graph Y1=sin(X+{,3,5}) and expects to see three graphs. However, the student sees only one graph displayed in the graph window. Has the calculator displayed all three graphs? Explain.

Chapter 2 Summary 2.1 Angles and Arcs • An angle is in standard position when its initial side is along the positive x-axis and its vertex is at the origin of the coordinate axes.

sin u 苷

opp hyp

cos u 苷

adj hyp

tan u 苷

opp adj

csc u 苷

hyp opp

sec u 苷

hyp adj

cot u 苷

adj opp

• Angle a is an acute angle when 0 a 90 ; it is an obtuse angle when 90 a 180 .

2.3 Trigonometric Functions of Any Angle

• a and b are complementary angles when a b 苷 90 ; they are supplementary angles when a b 苷 180 .

• Let P共x, y兲 be a point, except the origin, on the terminal side of an angle u in standard position. The six trigonometric functions of u are

• One radian is the measure of a central angle subtended by an arc of length r on a circle of radius r.

sin u 苷

y r

csc u 苷

r , y

y苷0

• The length of the arc s that subtends the central angle u (in radians) on a circle of radius r is given by s 苷 ru.

cos u 苷

x r

sec u 苷

r , x

x苷0

tan u 苷

y , x

cot u 苷

x , y

y苷0

• A point moves on a circular path with radius r at a constant s rate of u radians per unit of time t. Its linear speed is v 苷 , t u and its angular speed is v 苷 . t

2.2 Right Triangle Trigonometry • Let u be an acute angle of a right triangle. The six trigonometric functions of u are given by

x苷0

• Given ⬔u in standard position, its reference angle u is the smallest positive angle formed by the terminal side of ⬔u and the x-axis.

2.4 Trigonometric Functions of Real Numbers • The wrapping function pairs a real number with a point on the unit circle.

211

Summary • Let W be the wrapping function, t be a real number, and W共t兲苷 P共x, y兲. Then the trigonometric functions of the real number t are defined as follows: sin t 苷 y

csc t 苷

1 , y

y苷0

cos t 苷 x

sec t 苷

1 , x

x苷0

y tan t 苷 , x

x cot t 苷 , y

x苷0

• The graphs of y 苷 a csc bx and y 苷 a sec bx both have a period 2p of . The graphs of one period of each function, with a 0 兩b兩 and b 0, are shown below. y 5a

y 5a 3a

y苷0

3π 2b

a −a

• sin t, csc t, tan t, and cot t are odd functions. • cos t and sec t are even functions. • sin t, cos t, sec t, and csc t have period 2p.

3a

y = a csc bx

π 2b

π b

y = a sec bx π b

a 2π x b

−π

2b

−a

−3a

−3a

−5a

−5a

π 2b

3π x 2b

• tan t and cot t have period p.

2.7 Graphing Techniques • The domain and range of each trigonometric function is given in Table 2.6 on page 160. • An identity is an equation that is true for every number in the domain of the equation. The Fundamental Trigonometric Identities are given on page 163.

2.5 Graphs of the Sine and Cosine Functions • The graphs of y 苷 a sin bx and y 苷 a cos bx both have 2p an amplitude of 兩a兩 and a period of . The graph of each 兩b兩 function for a 0 and b 0 is shown below. y

• If y 苷 f (bx c), where f is a trigonometric function, then the graph of y 苷 f (bx c) can be produced by shifting the graph c c of y 苷 f (bx) horizontally units. The value is called b b the phase shift of y f(bx c). • Addition of ordinates is a method of graphing the sum of two functions by geometrically adding the values of their y-coordinates. • The factor g共x兲 in f 共x兲苷 g共x兲 cos x is called a damping factor. The graph of f lies on or between the graphs of the equations y 苷 g共x兲 and y 苷 g共x兲.

y

y = a sin bx

2.8 Harmonic Motion—An Application of the Sine and Cosine Functions

y = a cos bx a

a

2π b π π 2b b

3π 2b

2π b

x

π 2b

π b

• The equations of simple harmonic motion are x

3π 2b

and

y 苷 a sin 2p ft

where 兩a兩 is the amplitude, f is the frequency, y is the displacement, and t is the time.

−a

−a

y 苷 a cos 2p ft

• Functions of the form f 共t兲苷 aekt cos vt are used to model some forms of damped harmonic motion.

2.6 Graphs of the Other Trigonometric Functions • The graphs of y 苷 a tan bx and y 苷 a cot bx both have a p period of . The graph of one period of each function for 兩b兩 a 0 and b 0 is shown below. y 5a y = a tan bx 3a −π

4b

−π

2b

y 5a y = a cot bx 3a

−a

3π 4b

a

a π 4b

π x 2b

−a

−3a

−3a

−5a

−5a

π 4b

π 2b

π x b

212

Chapter 2

Trigonometric Functions

Chapter 2 Assessing Concepts 1. True or False: In the formula s 苷 ru, the measure of angle u must be in radians. 2. True or False: sec 2 u tan2 u 苷 1 is an identity. 3. True or False: The measure of one radian differs depending on the radius of the circle used.

冉冊

6. What is the point defined by W

7. What is the period of the graph of y 苷

»

3p 1 cos x? 2 4

8. Explain how to use the graph of y1 苷 sin

4. True or False: The graph of y 苷 sin x is symmetric with respect to the origin. 5. What is the measure of the reference angle for the angle 3p ? u苷 4

p ? 2

graph of y2 苷 sin

冉

冊

x p . 2 4

x to produce the 2

9. What is the domain of the function y 苷 cot x? 10. Consider the graph of y sec x on the interval [0, 2]. What are the equations of the vertical asymptotes of the graph?

Chapter 2 Review Exercises

11. Find the complement and supplement of the angle u whose measure is 65 .

13. Find the exact value of

3p b. tan 4

c. cot共225 兲

d. cos

2. Find the measure of the reference angle u for the angle u whose measure is 980 . 3. Convert 2 radians to the nearest hundredth of a degree. 14.

4. Convert 315 to radian measure. 5. Find the length (to the nearest hundredth of a meter) of the arc on a circle of radius 3 meters that subtends an angle of 75 . 6. Find the radian measure of the angle subtended by an arc of length 12 centimeters on a circle whose radius is 40 centimeters. 7. A car with a 16-inch-radius wheel is moving with a speed of 50 mph. Find the angular speed (to the nearest radian per second) of the wheel.

8. cos u

9. cot u

10. sin u

11. sec u

12. Find the values of the six trigonometric functions of an angle in standard position with the point P共1, 3兲 on the terminal side of the angle.

2p 3

Find the value of each of the following to the nearest ten-thousandth. a. cos 123

b. cot 4.22

c. sec 612

d. tan

2p 5

兹3 , 180 f 270 , find the exact 2

15. Given cos f 苷 value of a. sin f

b. tan f

16. Given tan f 苷 In Exercises 8 to 11, let be an acute angle of a right triangle 3 and csc . Evaluate each function. 2

冉冊冉冊

a. sec 150

兹3 , 90 f 180 , find the exact 3

value of a. sec f 17. Given sin f 苷

b. csc f 兹2 , 270 f 360 , find the exact 2

value of a. cos f

b. cot f

213

Review Exercises 18. Let W be the wrapping function. Evaluate

冉冊

p b. W 3

a. W共p兲

冉冊

冉

44. y 苷 2 csc 2x

5p c. W 4

d. W共28p兲

p 3

冊

46. y 苷 3 sin 2x 3

19. Is the function defined by f 共x兲苷 sin共x兲 tan共x兲 even, odd, or neither?

冉

48. y 苷 cos 3x

冉冊

45. y 苷 3 sec x

p 4

47. y 苷 2 cos 3x 3 p 2

冊

2

冉

49. y 苷 3 sin 4x

冊

2p 3 3

In Exercises 20 and 21, use the unit circle to show that each equation is an identity.

50. y 苷 2 sin 2x

20. cos共p t兲苷 cos t

52. A car climbs a hill that has a constant angle of 4.5 for a distance of 1.14 miles. What is the car’s increase in altitude?

21. tan共t兲苷 tan t

In Exercises 22 to 27, use trigonometric identities to write each expression in terms of a single trigonometric function or as a constant.

22. 1

24.

sin2 f cos2 f

23.

cos2 f sin2 f csc f

26. 1

tan f 1 cot f 1

27.

cos2 f 1 1 sin2 f

In Exercises 28 to 33, state the amplitude (if it exists), period, and phase shift of the graph of each function.

28. y 苷 3 cos共2x p兲

冉冉

29. y 苷 2 tan 3x

冊冊

30. y 苷 2 sin 3x

p 3

32. y 苷 4 sec 4x

3p 2

冉冊冉冊

31. y 苷 cos 2x

2p 2 3

33. y 苷 2 csc x

p 4

36. y 苷 2 sin

18°

38. y 苷

冉

40. y 苷 tan

x 2

冊

57. A mass of 5 kilograms is in equilibrium suspended from a spring. The mass is pulled down 0.5 foot and released. Find the period, frequency, and amplitude of the motion, assuming the mass oscillates in simple harmonic motion. Write an equation of motion. Assume k 苷 20.

冉冊 p 2

39. y 苷 3 cos 3共x p兲

58.

p 2

冉

43. y 苷 cot 2x

Graph the damped harmonic motion that is modeled by f 共t兲苷 3e0.75t cos pt

41. y 苷 2 cot 2x

冉冊

42. y 苷 tan x

2x 3

37. y 苷 cos x

1 p sin 2x 2 4

37°

80 ft

56. Find the amplitude, period, and frequency of the simple harmonic motion given by y 苷 2.5 sin 50t.

35. y 苷 sin

3x 2

55. HEIGHT OF A BUILDING Find the height of a building if the angle of elevation to the top of the building changes from 18 to 37 as an observer moves a distance of 80 feet toward the building.

3

In Exercises 34 to 51, graph each function.

34. y 苷 2 cos px

53. HEIGHT OF A TREE A tree casts a shadow of 8.55 feet when the angle of elevation of the sun is 55.3 . Find the height of the tree. 54. LINEAR SPEEDS ON A CAROUSEL A carousel has two circular rings of horses. The inner ring has a radius of 14.5 feet and the outer ring has a radius of 21.0 feet. The carousel makes one complete revolution every 24 seconds. How much greater, in feet per second, is the linear speed of a horse in the outer ring than the linear speed of a horse in the inner ring? Round to the nearest tenth of a foot per second.

25. sin2 f共tan2 f 1兲

1 tan2 f

51. y 苷 sin x 兹3 cos x

p 4

冊

where t is in seconds. Use the graph to determine how long (to the nearest tenth of a second) it will be until the absolute value of the displacement of the mass is always less than 0.01.

214

Chapter 2

Trigonometric Functions

» » » Quantitative Reasoning: Find the Periods of Trigonometric »»» Functions and Combined Musical Sound Tracks QR1. Let f be a function with period j and let g be a function with period k. The function f g is a periodic function provided there exist natural numbers j m 苷 . Furthermore, if m and n have no common m and n such that k n prime factors, then the period of f g is n . j 苷 m . k. For each of the following, determine the period of f g. a. f (x) 苷 sin x, g(x) 苷 cos

2 x 3

b. f(x) cos 3x, g(x) 苷 sin

c. f (x) 苷 tan 2x, g(x) 苷 sin 3x e. f (x) 苷 sec

4p 2p x, g(x) 苷 cot x 5 3

1 x 2

d. f (x) 苷 cot

3 2 x, g(x) 苷 cos x 3 4

f. f (x) 苷 csc

5 1 x, g(x) 苷 tan x 2 4

QR2. The following figure shows a guitar sound track with a period of 3 seconds and an electric bass track with a period of 2.5 seconds. Use the procedure in Exercise QR1 to find the period of the music produced when the two sound tracks are played together. 1

Tracks

2

3

4

5

6

7

8

Guitar

Electric Bass

QR3. A tambourine sound track has a period of 1.25 seconds and a drum sound track has a period of 2.25 seconds. Use the procedure in Exercise QR1 to find the period of the music produced when the two sound tracks are played together. QR4. An alto sax sound track has a period of 6 seconds, a piano sound track has a period of 4.5 seconds, and an electric guitar sound track has a period of 27 seconds. Find the period of the music produced when the three sound tracks are played together.

»

Chapter 2 Test

1. Convert 150 to exact radian measure. 2. Find the supplement of the angle whose radian measure 11 is p. Express your answer in terms of p. 12 3. Find the length (to the nearest tenth of a centimeter) of an arc that subtends a central angle of 75 in a circle of radius 10 centimeters.

4. ANGULAR SPEED A wheel is rotating at 6 revolutions per second. Find the angular speed in radians per second. 5. LINEAR SPEED A wheel with a diameter of 16 centimeters is rotating at 10 radians per second. Find the linear speed (in centimeters per second) of a point on the edge of the wheel. 3 6. If u is an acute angle of a right triangle and tan u 苷 , 7 find sec u.

Cumulative Review Exercises 7. Use a calculator to find the value of csc 67 to the nearest ten-thousandth. 8. Find the exact value of tan

p p p cos sin . 6 3 2

冉冊

9. Find the exact coordinates of W

11p . 6

sec 2 t 1 10. Express in terms of a single trigonometric sec 2 t function. 11. State the period of y 苷 4 tan 3x.

冉

冊

12. State the amplitude, period, and phase shift for the funcp tion y 苷 3 cos 2x . 2

冉

冊

13. State the period and phase shift for the function p p y 苷 2 cot x . 3 6

14. Graph one full period of y 苷 3 cos

1 x. 2

15. Graph one full period of y 苷 2 sec

冉

215

1 x. 2

冊

16. Write a sentence that explains how to obtain the graph of p 1 from the graph of y 苷 2 sin 2x. y 苷 2 sin 2x 2 17. Graph one full period of y 苷 2 sin

x . 2

18. Graph one full period of y 苷 sin x cos 2x. 19. HEIGHT OF A TREE The angle of elevation from point A to the top of a tree is 42.2 . From point B, which is 5.24 meters from A and on a line through the base of the tree and A, the angle of elevation to the top of the tree is 37.4 . Find the height of the tree. 20. Write the equation for simple harmonic motion given that the amplitude is 13 feet, the period is 5 seconds, and the displacement is zero when t 苷 0.

Cumulative Review Exercises 1. Find the distance between the points P共3, 2兲 and Q共4, 1兲. 11. Convert 2. The hypotenuse of a right triangle has a length of 1 and 1 one of its legs has a length of . Find the length of the 2 other leg.

5 to degrees. 4

冉冊

12. Evaluate f 共x兲苷 sin x

6

13. Evaluate f 共x兲苷 sin x sin 3. Find the x- and the y-intercept(s) of the graph of f 共x兲苷 x2 9. 4. Determine whether f 共x兲苷

x is an even function or x2 1

2 6. Use interval notation to state the domain of f 共x兲苷 . x4 7. Solve: x2 x 6 苷 0 8. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共x 3兲. 9. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共x兲. 10. Convert 300° to radians.

. 3

for x 苷 . 6 3

14. Find the exact value of cos2 45° sin2 60°. 15. Determine the sign of tan given that

an odd function. x 5. Find the inverse of f 共x兲苷 . 2x 3

for x 苷

. 2

16. What is the measure of the reference angle for the angle 苷 210°? 17. What is the measure of the reference angle for the angle 2 苷 ? 3 18. Use interval notation to state the domain of f 共x兲苷 sin x, where x is a real number. 19. Use interval notation to state the range of f 共x兲苷 cos x, where x is a real number. 20. If is an acute angle of a right triangle and tan 苷 find sin .

3 , 4

3

Trigonometric Identities and Equations 3.1 Verification of Trigonometric Identities 3.2 Sum, Difference, and Cofunction Identities 3.3 Double- and Half-Angle Identities 3.4 Identities Involving the Sum of Trigonometric Functions 3.5 Inverse Trigonometric Functions 3.6 Trigonometric Equations Trigonometric Equations and Applications This chapter explores trigonometric identities, inverse trigonometric functions, and trigonometric equations. Understanding these concepts will provide you with a foundation for more advanced mathematical studies. You will also gain additional problem-solving skills and learn procedures that can be used to solve a wide variety of real-life applications. For example, in Exercise 83 on page 266, you will determine the minimum altitude a communications satellite must attain to provide service over a desired region. In Quantitative Reasoning Exercise 2a on page 289, you will find the launch angle that minimizes the velocity at which a basketball must be launched to make a shot from a given distance.

α

β s

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

216

3.1

Section 3.1 ■

■

Fundamental Trigonometric Identities Verification of Trigonometric Identities

217

Verification of Trigonometric Identities

Verification of Trigonometric Identities Fundamental Trigonometric Identities The domain of an equation consists of all values of the variable for which every term is defined. For example, the domain of sin x cos x 苷 cos x sin x

(1)

includes all real numbers x except x 苷 kp, where k is an integer, because sin x 苷 0 for x 苷 kp, and division by 0 is undefined. An identity is an equation that is true for all of its domain values. Table 3.1 lists identities that were introduced earlier. Table 3.1

Fundamental Trigonometric Identities

Reciprocal identities

sin x 苷

1 csc x

cos x 苷

1 sec x

Ratio identities

tan x 苷

sin x cos x

cot x 苷

cos x sin x

Pythagorean identities

sin2 x cos2 x 苷 1

tan2 x 1 苷 sec2 x

1 cot 2 x 苷 csc2 x

Odd–even identities

sin共x兲苷 sin x cos共x兲苷 cos x

tan共x兲苷 tan x cot共x兲苷 cot x

sec共x兲苷 sec x csc共x兲苷 csc x

tan x 苷

1 cot x

Verification of Trigonometric Identities To verify an identity, we show that one side of the identity can be rewritten in an equivalent form that is identical to the other side. There is no one method that can be used to verify every identity; however, the following guidelines should prove useful. Guidelines for Verifying Trigonometric Identities ■

If one side of the identity is more complex than the other, then it is generally best to try first to simplify the more complex side until it becomes identical to the other side.

■

Perform indicated operations such as adding fractions or squaring a binomial. Also be aware of any factorization that may help you to achieve your goal of producing the expression on the other side.

■

Make use of previously established identities that enable you to rewrite one side of the identity in an equivalent form.

■

Rewrite one side of the identity so that it involves only sines and/or cosines. Continued 䉴

218

Chapter 3

Trigonometric Identities and Equations

■

Rewrite one side of the identity in terms of a single trigonometric function.

■

Multiplying both the numerator and the denominator of a fraction by the same factor (such as the conjugate of the denominator or the conjugate of the numerator) may get you closer to your goal.

■

Keep your goal in mind. Does it involve products, quotients, sums, radicals, or powers? Knowing exactly what your goal is may provide the insight you need to verify the identity.

In Example 1 we verify an identity by rewriting the more complicated side so that it involves only sines and cosines. EXAMPLE 1

»

Change to Sines and Cosines to Verify an Identity

Verify the identity sin x cot x sec x 苷 1.

Solution The left side of the identity is more complicated than the right side. We will try to verify the identity by rewriting the left side so that it involves only sines and cosines. sin x cot x sec x 苷 sin x

cos x 1 sin x cos x

sin x cos x sin x cos x 苷1 苷

• Apply the fundamental identities 1 cos x cot x and sec x . sin x cos x • Multiply the fractions to produce a single fraction. • Simplify by dividing out the common factors.

We have rewritten the left side of the equation so that it is identical to the right side. Thus we have verified that the equation is an identity.

»

Try Exercise 2, page 222

QUESTION

Is cos共x兲苷 cos x an identity?

EXAMPLE 2

»

Use a Pythagorean Identity to Verify an Identity

Verify the identity 1 2 sin2 x 苷 2 cos2 x 1.

ANSWER

Yes, cos共x兲苷 cos x is one of the odd–even identities shown in Table 3.1.

3.1

take note Each of the Pythagorean identities

Solution Rewrite the right side of the equation. 2 cos2 x 1 苷 2共1 sin2 x兲 1 苷 2 2 sin2 x 1 苷 1 2 sin2 x

can be written in several different forms. For instance, sin2 x cos2 x 苷 1 also can be written as sin2 x 苷 1 cos2 x and as cos2 x 苷 1 sin2 x

219

Verification of Trigonometric Identities

• cos2 x 1 sin2 x • Simplify.

»

Try Exercise 12, page 223

Figure 3.1 shows the graph of f共x兲苷 1 2 sin2 x and the graph of g共x兲苷 2 cos2 x 1 on the same coordinate axes. The fact that the graphs appear to be identical on the interval 关2p, 2p兴 supports the verification in Example 2.

Plot1 Plot2 Plot3

\Y1 = 1 – 2(sin(x))2 \Y2 = 2(cos(x))2 – 1 1.5

−2π

2π

−1.5

Y1 = Y2

Figure 3.1

EXAMPLE 3

»

Factor to Verify an Identity

Verify the identity csc2 x cos2 x csc2 x 苷 1.

Solution Simplify the left side of the equation. csc2 x cos2 x csc2 x 苷 csc2 x共1 cos2 x兲苷 csc2 x sin2 x 1 苷 sin2 x 苷 1 sin2 x

• Factor out csc2 x. • 1 cos2 x sin2 x • csc2 x

1 sin2 x

»

Try Exercise 24, page 223

In the next example we make use of the guideline that states that it may be helpful to multiply both the numerator and the denominator of a fraction by the same factor.

EXAMPLE 4

»

Verify the identity

Multiply by a Conjugate to Verify an Identity

sin x 1 cos x 苷 . 1 cos x sin x

Continued 䉴

220

Chapter 3

take note The sum a b and the difference a b are called conjugates of

Trigonometric Identities and Equations Solution Multiply the numerator and denominator of the left side of the identity by the conjugate of 1 cos x, which is 1 cos x. sin x sin x 1 cos x 苷 1 cos x 1 cos x 1 cos x

each other.

sin x共1 cos x兲 1 cos2 x sin x共1 cos x兲 1 cos x 苷苷 sin2 x sin x 苷

»

Try Exercise 34, page 223

In Example 5 we verify an identity by first rewriting the more complicated side so that it involves only sines and cosines. After further algebraic simplification we are able to establish that the equation is an identity. EXAMPLE 5

»

Verify the identity

Change to Sines and Cosines to Verify an Identity

sin x tan x 苷 tan x. 1 cos x

Solution The left side of the identity is more complicated than the right side. We will try to verify the identity by rewriting the left side so that it involves only sines and cosines. sin x cos x 1 cos x sin x cos x sin x cos x cos x 苷 1 cos x

sin x tan x 苷 1 cos x

sin x

sin x cos x sin x cos x 苷 1 cos x sin x cos x sin x 1 cos x 苷 cos x 1 sin x cos x sin x 1 苷 cos x 1 cos x 1 sin x 共cos x 1兲苷 cos x 1 cos x

• Use the identity sin x tan x . cos x • The common denominator for the terms in the numerator is cos x. Rewrite the numerator so that each term has a denominator of cos x. • Add the terms in the numerator. • Rewrite the complex fraction as a division. • Invert and multiply. • Factor.

3.1

Verification of Trigonometric Identities

sin x 共1 cos x兲 1 cos x 1 cos x sin x 苷 cos x 苷 tan x

苷

221

• Simplify.

• Use the identity

sin x tan x . cos x

We have rewritten the left side of the equation so that it is identical to the right side. Thus we have verified that the equation is an identity.

»

Try Exercise 44, page 224

In the previous examples we verified trigonometric identities by rewriting one side of the identity in equivalent forms until that side appeared identical to the other side. A second method is to work with each side separately to produce a trigonometric expression that is equivalent to both sides. In the following paragraphs we use this procedure to verify the identity 1 cos x 苷共csc x cot x兲2 1 cos x Working with the left side gives us 1 cos x 1 cos x 1 cos x 苷 1 cos x 1 cos x 1 cos x

(1)

• Multiply both numerator and denominator by the conjugate of the denominator.

苷

1 2 cos x cos2 x 1 cos2 x

(2)

• Multiply and rewrite as a single fraction.

苷

1 2 cos x cos2 x sin2 x

(3)

• Use the identity 1 cos2 x sin2 x.

When you reach a point at which you are unsure how to proceed, stop and see what results can be obtained by working with the right side. 共csc x cot x兲2 苷 csc 2 x 2 csc x cot x cot 2 x (4) • Square the binomial. 2 1 1 cos x cos x 苷 2 (5) • Rewrite using fundasin2 x sin x sin x sin2 x mental identities.

冉冊冉冊

苷

1 2 cos x cos2 x sin2 x

(6)

• Add the fractions.

At this point we have verified the identity because we have shown that each side is equal to the same trigonometric expression. It is worth noting that if we arranged the above equations in the order (1), (2), (3), (5), and (4), we would have a verification that starts with the left side of the original identity and produces the right side. When we verify an identity, we are not allowed to perform the same mathematical operation on both sides of the equation because this can lead to an incorrect result. For instance, consider the equation tan x 苷 tan x, which is not an identity. If we square both sides of the equation, we produce the identity tan2 x 苷 tan2 x. We can avoid the problem of converting a non-identity into an identity and vice versa by using only mathematical operations and procedures that are reversible. We can see that the operation of squaring is not a reversible operation because it is not possible to take square roots (reverse the squaring operation) and convert tan2 x 苷 tan2 x back into tan x 苷 tan x.

222

Chapter 3

Trigonometric Identities and Equations There is often more than one way to verify a trigonometric identity. Some of these approaches are shorter and considered more elegant than others. As a beginner, you should be content with finding a verification. After you have had lots of practice, you may decide to seek out different verifications and compare them to see which you think is most efficient.

Topics for Discussion 1. Explain why tan 苷

sin is not an identity. cos

2. Is cos 兩x兩苷兩cos x兩 an identity? Explain. What about cos 兩x兩苷 cos x? Explain. 3. The identity sin2 x cos2 x 苷 1 is one of the Pythagorean identities. What are the other two Pythagorean identities, and how are they derived? 4. The graph of y 苷 sin of y 苷

冑

x for 0 x 2p is shown on the left below. The graph 2

1 cos x for 0 x 2p is shown on the right below. 2

1

1

2π

0

−1

2π

0

−1

y 苷 sin

x 2

y苷

冑

1 cos x 2

The graphs appear identical, but the equation sin

x 苷 2

冑

1 cos x 2

is not an identity. Explain.

Exercise Set 3.1 In Exercises 1 to 56, verify each identity.

1. tan x csc x cos x 苷 1

5. 共sin x cos x兲共sin x cos x兲苷 1 2 cos2 x 6. 共tan x兲共1 cot x兲苷 tan x 1

2. tan x sec x sin x 苷 tan x 2

4 sin x 1 苷 2 sin x 1 2 sin x 1

7.

1 cos x sin x 1 苷 sin x cos x sin x cos x

8.

1 3 cos x 3 sin x 苷 sin x cos x sin x cos x

2

3.

4.

sin2 x 2 sin x 1 苷 sin x 1 sin x 1

3.1 9.

cos x 苷 sec x tan x 1 sin x

sin x 苷 csc x cot x 10. 1 cos x 11.

1 tan4 x 苷 1 tan2 x sec2 x

Verification of Trigonometric Identities 27. sin2 x cos2 x 苷 2 sin2 x 1 28. sin2 x cos2 x 苷 1 2 cos2 x 29.

1 1 苷 csc 2 x sec 2 x sin2 x cos2 x

30.

1 1 苷 csc 2 x sec 2 x tan2 x cot 2 x

12. sin4 x cos4 x 苷 sin2 x cos2 x 13.

1 tan3 x 苷 1 tan x tan2 x 1 tan x

14.

cos x tan x sin x 苷0 cot x 1 sin x sin x 1 苷 1 sin x 1 sin x sin x

sin x 2 15.

16.

sin x sin x 苷 2 cot x 1 cos x 1 cos x

31. sec x cos x 苷 sin x tan x 32. tan x cot x 苷 sec x csc x 1 1 sin x 苷 tan2 x 2 tan x sec x sec 2 x 33. 1 1 sin x 1 1 sin x cos x cos2 x sin2 x 苷 34. 1 1 1 2 cos x sin x sin x cos x

17. 共sin x cos x兲2 苷 1 2 sin x cos x

35. sin4 x cos4 x 苷 2 sin2 x 1

18. 共tan x 1兲2 苷 sec 2 x 2 tan x

36. sin6 x cos6 x 苷 sin4 x sin2 x cos2 x cos4 x 1 1 cos x 苷 1 cos x sin2 x

19.

cos x 苷 sec x tan x 1 sin x

37.

20.

sin x 苷 csc x cot x 1 cos x

38. 1 sin x 苷

cos2 x 1 sin x

21. csc x 苷

cot x tan x sec x

39.

sin x cos x 1 cot x 苷 1 sin x 1 sin x csc x 1

22. sec x 苷

cot x tan x csc x

40.

tan x cot x 苷 1 cot x 1 tan x 1 tan x

23.

cos x tan x 2 cos x tan x 2 苷 cos x 1 tan x 2

41.

1 1 苷 2 cot x csc x 1 cos x 1 cos x

24.

2 sin x cot x sin x 4 cot x 2 苷 sin x 2 2 cot x 1

42.

1 1 苷 2 tan x sec x 1 sin x 1 sin x

25. sec x tan x 苷

1 sin x cos x

26. cot x csc x 苷

cos x 1 sin x

1 csc x sin x 2 苷 43. 1 cos2 x sin x sin x

223

224

Chapter 3

Trigonometric Identities and Equations

44.

2 cot x 苷 2 cos2 x cot x tan x

45.

1 csc x cot x 苷 2 sec x 1 csc x cot x

46. sec2 x csc2 x 苷

47.

48.

49.

50.

51.

52.

In Exercises 57 to 64, compare the graphs of each side of the equation to predict whether the equation is an identity.

57. sin 2x 苷 2 sin x cos x

冉冊

59. sin x cos x 苷兹2 sin x

tan x cot x sin x cos x

冑

1 sin x 1 sin x 苷 , 1 sin x cos x

冉冉冉冉

cos x 0

sin3 x cos3 x 苷 1 sin x cos x sin x cos x 1 sin x 1 sin x 苷 4 sec x tan x 1 sin x 1 sin x sec x 1 sec x 1 苷 4 csc x cot x sec x 1 sec x 1

p 3

62. cos x

p 4

63. sin x

p 6

64. sin x

p 3

冊冊冊冊

苷 cos x cos

p p sin x sin 3 3

苷 cos x cos

p p sin x sin 4 4

苷 sin x sin

p p cos x cos 6 6

苷 sin x cos

p p cos x sin 3 3

65. 共sin x cos x兲2 苷 sin2 x cos2 x

cos x 1 sin x 53. 苷0 cos x 1 sin x

66. tan 2x 苷 2 tan x

54. 共sin x cos x 1兲2 苷 2共sin x 1兲共cos x 1兲

67. cos共x 30 兲苷 cos x cos 30 68. 兹1 sin2 x 苷 cos x

55.

sec x tan x 共sin x 1兲2 苷 sec x tan x cos 2 x

56.

sin x cos x csc x cot x 2 cos x 苷 sin x cos x 1 cot 2 x 2

61. cos x

In Exercises 65 to 70, verify that the equation is not an identity by finding an x value for which the left side of the equation is not equal to the right side.

cos x 1 苷 2 csc 2 x 1 1 cos x 1 cos x

3

p 4

60. cos 2x 苷 2 cos2 x 1

cos x cot x sin x 苷 2 sin x cot x

3

58. sin2 x cos2 x 苷 1

69. tan4 x sec4 x 苷 tan2 x sec 2 x 2

70. 兹1 tan2 x 苷 sec x

Connecting Concepts In Exercises 71 to 76, verify the identity.

71.

cos x 1 sin x cos x 苷 1 sin x cos x sin x 1

72.

1 tan x sec x 1 sec x 苷 1 tan x sec x tan x

73.

74.

2 sin4 x 2 sin2 x cos 2 x 3 sin2 x 3 cos2 x 2 sin2 x 3 苷 1 csc 2 x 2 4 tan x sec 2 x 4 tan x sec 2 x 1 苷1 4 tan3 x tan2 x

3.2

75.

76.

Sum, Difference, and Cofunction Identities

sin x共tan x 1兲 2 tan x cos x 苷 tan x sin x cos x

77. Verify the identity sin4 x cos4 x 苷 1 2 sin2 x cos2 x by completing the square of the left side of the identity.

sin2 x cos x cos3 x sin3 x cos x sin x cos3 x 1 sin2 x cos x 苷 1 sin x

78. Verify the identity tan4 x sec4 x 苷 1 2 tan2 x sec 2 x by completing the square of the left side of the identity.

Projects 1.

GRADING A QUIZ Suppose that you are a teacher’s assistant. You are to assist the teacher of a trigonometry class by grading a four-question quiz. Each question asks the student to find a trigonometric expression that models a given application. The teacher has prepared an answer key. These answers are shown in the next column. A student gives as answers the expressions shown in the far right column. Determine for which problems the student has given a correct response.

Section 3.2 ■

Identities That Involve ( )

■ ■

■

225

Cofunctions Additional Sum and Difference Identities Reduction Formulas

Answer Key 1. csc x sec x 2. cos2 x 3. cos x cot x 4. csc x cot x

Student’s Response 1. cot x tan x 2. 共1 sin x兲共1 sin x兲 3. csc x sec x 4. sin x共cot x cot 3 x兲

Sum, Difference, and Cofunction Identities P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A16.

PS1. Compare cos共a b兲 and cos a cos b sin a sin b for a 苷

p p and b 苷 . [2.2] 2 6

PS2. Compare sin共a b兲 and sin a cos b cos a sin b for a 苷

p p and b 苷 . [2.2] 2 3

PS3. Compare sin共90 u兲 and cos u for u 苷 30 , u 苷 45 , and u 苷 120 . [2.2]

冉冊

PS4. Compare tan

p p p 4p u and cot u for u 苷 ,u苷 , and u 苷 . [2.2] 2 6 4 3

PS5. Compare tan共a b兲 and

tan a tan b p p for a 苷 and b 苷 . [2.2] 1 tan a tan b 3 6

PS6. Find the value of sin关共2k 1兲p兴, where k is any arbitrary integer. [2.2]

226

Chapter 3

Trigonometric Identities and Equations

Identities That Involve ( ) Each identity in Section 3.1 involved only one variable. We now consider identities that involve a trigonometric function of the sum or difference of two variables.

Sum and Difference Identities cos共a b兲苷 cos a cos b sin a sin b cos共a b兲苷 cos a cos b sin a sin b sin共a b兲苷 sin a cos b cos a sin b sin共a b兲苷 sin a cos b cos a sin b tan共a b兲苷

tan a tan b 1 tan a tan b

tan共a b兲苷

tan a tan b 1 tan a tan b

Proof To establish the identity for cos共a b兲, we make use of the unit circle shown in Figure 3.2. The angles a and b are drawn in standard position, with OA and OB as the terminal sides of a and b, respectively. The coordinates of A are 共cos a, sin a兲, and the coordinates of B are 共cos b, sin b兲. The angle 共a b兲 is formed by the terminal sides of the angles a and b (angle AOB).

y

C(cos (α − β ), sin (α − β ) ) B(cos β, sin β ) α −β

β α

α −β

O

D(1, 0) x

A(cos α, sin α)

Figure 3.2

An angle equal in measure to angle 共a b兲 is placed in standard position in the same figure (angle COD). From geometry, if two central angles of a circle have the same measure, then their chords are also equal in measure. Thus the chords AB and CD are equal in length. Using the distance formula, we can calculate the lengths of the chords AB and CD. d共A, B兲苷兹共cos a cos b兲2 共sin a sin b兲2 d共C, D兲苷兹关cos共a b兲 1兴 2 关sin共a b兲 0兴 2

3.2

Sum, Difference, and Cofunction Identities

227

Because d共A, B兲苷 d共C, D兲, we have 兹共cos a cos b兲2 共sin a sin b兲2 苷兹关cos共a b兲 1兴 2 关sin共a b兲兴 2 Squaring each side of the equation and simplifying, we obtain 共cos a cos b兲2 共sin a sin b兲2 苷关cos共a b兲 1兴 2 关sin共a b兲兴 2 cos2 a 2 cos a cos b cos2 b sin2 a 2 sin a sin b sin2 b 苷 cos2共a b兲 2 cos共a b兲 1 sin2共a b兲 cos2 a sin2 a cos2 b sin2 b 2 cos a cos b 2 sin a sin b 苷 cos2共a b兲 sin2共a b兲 1 2 cos共a b兲 Simplifying by using sin2 u cos2 u 苷 1, we have 2 2 sin a sin b 2 cos a cos b 苷 2 2 cos共a b兲 Solving for cos共a b兲 gives us cos共a b兲苷 cos a cos b sin a sin b

◆

To derive an identity for cos共a b兲, write cos共a b兲 as cos关a 共b兲兴. cos共a b兲苷 cos关a 共b兲兴苷 cos a cos共b兲 sin a sin共b兲 Recall that cos共b兲苷 cos b and sin共b兲苷 sin b. Substituting into the previous equation, we obtain the identity cos共a b兲苷 cos a cos b sin a sin b EXAMPLE 1

»

Evaluate a Trigonometric Expression

Use an identity to find the exact value of cos共60 45 兲.

Solution Use the identity cos共a b兲苷 cos a cos b sin a sin b with a 苷 60 and b 苷 45 . cos共60 45 兲苷 cos 60 cos 45 sin 60 sin 45 苷

冉冊冉冊冉冊冉冊

苷

兹2 兹6 4 4

苷

兹2 兹6 4

1 2

兹2 2

兹3 2

兹2 2

• Substitute. • Evaluate each factor.

• Simplify.

»

Try Exercise 4, page 233

Cofunctions Any pair of trigonometric functions f and g for which f共x兲苷 g共90 x兲 and g共x兲苷 f共90 x兲 are said to be cofunctions.

228

Chapter 3

Trigonometric Identities and Equations Cofunction Identities

take note To visualize the cofunction

sin共90 u兲苷 cos u

cos共90 u兲苷 sin u

triangle shown in the following

tan共90 u兲苷 cot u

cot共90 u兲苷 tan u

figure.

sec共90 u兲苷 csc u

csc共90 u兲苷 sec u

identities, consider the right

c

90° − θ

b

If u is in radian measure, replace 90° with

p . 2

θ

a

If u is the degree measure of one of the acute angles, then the degree

To verify that the sine function and the cosine function are cofunctions, we make use of the identity for cos共a b兲. cos共90 b兲苷 cos 90 cos b sin 90 sin b 苷 0 cos b 1 sin b

measure of the other acute angle is 共90 u兲. Using the definitions of the trigonometric functions gives us sin u 苷

b 苷 cos 共90 u兲 c

b tan u 苷苷 cot 共90 u兲 a c sec u 苷苷 csc 共90 u兲 a

which gives cos共90 b兲苷 sin b Thus the sine of an angle is equal to the cosine of its complement. Using cos共90 b兲苷 sin b with b 苷 90 a, we have cos a 苷 cos关90 共90 a兲兴苷 sin共90 a兲 Therefore, cos a 苷 sin共90 a兲

These identities state that the

value of a trigonometric function of u is equal to the cofunction of the

We can use the ratio identities to show that the tangent and cotangent functions are cofunctions.

complement of u.

tan共90 u兲苷

sin共90 u兲 cos u 苷苷 cot u cos共90 u兲 sin u

cot共90 u兲苷

cos共90 u兲 sin u 苷苷 tan u sin共90 u兲 cos u

The secant and cosecant functions are also cofunctions.

EXAMPLE 2

»

Write an Equivalent Expression

Use a cofunction identity to write an equivalent expression for sin 20°.

Solution The value of a given trigonometric function of u, measured in degrees, is equal to its cofunction of 90 u. Thus sin 20 苷 cos共90 20 兲苷 cos 70

»

Try Exercise 20, page 233

3.2

229

Sum, Difference, and Cofunction Identities

Additional Sum and Difference Identities We can use the cofunction identities to verify the remaining sum and difference identities. To derive an identity for sin共a b兲, substitute a b for u in the cofunction identity sin u 苷 cos共90 u兲. sin u 苷 cos共90 u兲 sin共a b兲苷 cos关90 共a b兲兴 • Replace with . • Rewrite as the difference of two angles. 苷 cos关共90 a兲 b兴苷 cos共90 a兲 cos b sin共90 a兲 sin b 苷 sin a cos b cos a sin b Therefore, sin共a b兲苷 sin a cos b cos a sin b We also can derive an identity for sin共a b兲 by rewriting 共a b兲 as 关a 共b兲兴. sin共a b兲苷 sin关a 共b兲兴苷 sin a cos共b兲 cos a sin共b兲苷 sin a cos b cos a sin b

• cos() cos sin() sin

Thus sin共a b兲苷 sin a cos b cos a sin b The identity for tan共a b兲 is a result of the identity tan u 苷 identities for sin共a b兲 and cos共a b兲.

sin u and the cos u

sin共a b兲 sin a cos b cos a sin b 苷 cos共a b兲 cos a cos b sin a sin b sin a cos b cos a sin b • Multiply both the numerator and the denominator by cos a cos b cos a cos b 苷 1 cos a cos b sin a sin b and simplify. cos cos cos a cos b cos a cos b

tan 共a b兲苷

Therefore, tan共a b兲苷

tan a tan b 1 tan a tan b

The tangent function is an odd function, so tan共u兲苷 tan u. Rewriting 共a b兲 as 关a 共b兲兴 enables us to derive an identity for tan共a b兲. tan共a b兲苷 tan关a 共b兲兴苷

tan a tan共b兲 1 tan a tan共b兲

Therefore, tan共a b兲苷

tan a tan b 1 tan a tan b

230

Chapter 3

Trigonometric Identities and Equations The sum and difference identities can be used to simplify some trigonometric expressions. EXAMPLE 3

»

Simplify Trigonometric Expressions

Write each expression in terms of a single trigonometric function. sin 5x cos 3x cos 5x sin 3x

a.

b.

tan 4a tan a 1 tan 4a tan a

Solution a.

sin 5x cos 3x cos 5x sin 3x 苷 sin共5x 3x兲苷 sin 2x

b.

tan 4a tan a 苷 tan共4a a兲苷 tan 5a 1 tan 4a tan a

»

Try Exercise 26, page 233

EXAMPLE 4

»

Evaluate a Trigonometric Function

4 5 for a in Quadrant II and tan b 苷 for b in 3 12 Quadrant IV, find sin共a b兲. Given tan a 苷

Solution y 4 苷 and the terminal side of a is x 3 in Quadrant II, P1共3, 4兲 is a point on the terminal side of a. Similarly, P2共12, 5兲 is a point on the terminal side of b. See Figure 3.3. Because tan a 苷

y

y

4

P1(− 3, 4)

β

5

6

O

12

x

13

α

−4

O

4

x

−6

P2 (12, −5)

Figure 3.3

Using the Pythagorean Theorem, we find that the length of the line segment OP1 is 5 and the length of OP2 is 13. sin共a b兲苷 sin a cos b cos a sin b 4 12 3 5 48 15 63 苷苷苷 5 13 5 13 65 65 65

»

Try Exercise 38, page 233

3.2 EXAMPLE 5

Sum, Difference, and Cofunction Identities

»

231

Verify an Identity

Verify the identity cos 共p u兲苷 cos u.

Solution Plot1 Plot2 Plot3 \Y1 = cos(π–x) \Y2 = -cos(x)

cos共p u兲苷 cos p cos u sin p sin u 苷 1 cos u 0 sin u 苷 cos u

1.5

• Use the identity for cos( ).

»

Try Exercise 50, page 234

−2π

2π

Y1 = Y2

Figure 3.4 shows the graphs of Y1 苷 cos共p u兲 and Y2 苷 cos u on the same coordinate axes. The fact that the graphs appear to be identical supports the verification in Example 5.

−1.5

Figure 3.4

EXAMPLE 6

»

Verify the identity

Verify an Identity

cos 4u sin 4u cos 5u 苷 . sin u cos u sin u cos u

Solution Subtract the fractions on the left side of the equation. cos 4u sin 4u cos 4u cos u sin 4u sin u 苷 sin u cos u sin u cos u cos共4u u兲苷 sin u cos u 苷

• Use the identity for cos( ).

cos 5u sin u cos u

»

Try Exercise 62, page 234

Plot1 Plot2 Plot3

\Y1 = (cos(4x))/sin(x)–(sin(4x) )/cos(x) \Y2 = (cos(5x))/(sin(x)cos(x))

Figure 3.5 shows the graph of Y1 苷

7

cos 4u sin 4u and the graph of sin u cos u

cos 5u on the same coordinate axes. The fact that the graphs appear to sin u cos u be identical supports the verification in Example 6. Y2 苷

−2π

2π

Y1 = Y2 −7

Figure 3.5

232

Chapter 3

Trigonometric Identities and Equations

Reduction Formulas The sum or difference identities can be used to write expressions such as sin共u kp兲

sin共u 2kp兲

and

cos关u 共2k 1兲p兴

where k is an integer, as expressions involving only sin u or cos u. The resulting formulas are called reduction formulas.

EXAMPLE 7

»

Find Reduction Formulas

Write as a function involving only sin u. sin关u 共2k 1兲p兴,

where k is an integer

Solution Applying the identity sin共a b兲苷 sin a cos b cos a sin b yields sin关u 共2k 1兲p兴苷 sin u cos关共2k 1兲p兴 cos u sin关共2k 1兲p兴 If k is an integer, then 2k 1 is an odd integer. The cosine of any odd multiple of p equals 1, and the sine of any odd multiple of p is 0. This gives us sin关u 共2k 1兲p兴苷共sin u兲共1兲共cos u兲共0兲苷 sin u Thus sin关u 共2k 1兲p兴苷 sin u for any integer k.

»

Try Exercise 76, page 235

QUESTION

Is sin共u 2kp兲苷 sin u a reduction formula?

Topics for Discussion 1. Does sin共a b兲苷 sin a sin b for all values of a and b? If not, find nonzero values of a and b for which sin共a b兲苷 sin a sin b. 2. If k is an integer, then 2k 1 is an odd integer. Do you agree? Explain. 3. What are the trigonometric cofunction identities? Explain. 4. Is tan共u kp兲苷 tan u, where k is an integer, a reduction formula? Explain.

ANSWER

Yes. sin 共u 2kp兲苷 sin u cos 共2kp兲 cos u sin 共2kp兲苷共sin u兲共1兲共cos u兲共0兲苷 sin u.

3.2

Sum, Difference, and Cofunction Identities

233

Exercise Set 3.2 In Exercises 1 to 18, find (if possible) the exact value of the expression.

1. sin共45 30 兲

2. sin共330 45 兲

3. cos共45 30 兲

4. cos共120 45 兲

5. tan共45 30 兲

6. tan共240 45 兲

冉冉冉

7. sin

5p p 4 6

9. cos

p 3p 4 6

p p 11. tan 6 4

冊冊冊

冉冉冉

8. sin

4p p 3 4

10. cos

p p 4 3

冊冊冊

11p p 12. tan 6 4

13. cos 212 cos 122 sin 212 sin 122

26. sin x cos 3x cos x sin 3x 27. cos x cos 2x sin x sin 2x 28. cos 4x cos 2x sin 4x sin 2x 29. sin 7x cos 3x cos 7x sin 3x 30. cos x cos 5x sin x sin 5x 31. cos 4x cos共2x兲 sin 4x sin共2x兲 32. sin共x兲 cos 3x cos共x兲 sin 3x 33. sin

2x x 2x x cos cos sin 3 3 3 3

34. cos

x 3x x 3x cos sin sin 4 4 4 4

14. sin 167 cos 107 cos 167 sin 107 15. sin

16. cos

35.

tan 3x tan 4x 1 tan 3x tan 4x

36.

tan 2x tan 3x 1 tan 2x tan 3x

p 5p p 5p cos cos sin 12 4 12 4 p p p p cos sin sin 12 4 12 4

In Exercises 37 to 48, find the exact value of the given functions.

p 7p tan tan 12 4 17. 7p p 1 tan tan 12 4

4 15 , a in Quadrant II, and tan b 苷 , b 3 8 in Quadrant III, find

37. Given tan a 苷

a. sin共a b兲

p p tan 6 3 18. p p tan 1 tan 6 3 tan

c. tan共a b兲

24 8 , a in Quadrant I, and sin b 苷 , b in 7 17 Quadrant III, find

38. Given tan a 苷

In Exercises 19 to 24, use a cofunction identity to write an equivalent expression for the given value.

19. sin 42

20. cos 80

21. tan 15

22. cot 2

23. sec 25

24. csc 84

a. sin共a b兲

In Exercises 25 to 36, write each expression in terms of a single trigonometric function.

b. cos共a b兲

c. tan共a b兲

3 5 , a in Quadrant I, and cos b 苷 , b in 5 13 Quadrant II, find

39. Given sin a 苷

a. sin共a b兲

25. sin 7x cos 2x cos 7x sin 2x

b. cos共a b兲

b. cos共a b兲

c. tan共a b兲

4 24 , a in Quadrant II, and cos b 苷 , b in 25 5 Quadrant III, find

40. Given sin a 苷

a. cos共b a兲

b. sin共a b兲

c. tan共a b兲

234

Chapter 3

Trigonometric Identities and Equations

4 12 , a in Quadrant III, and cos b 苷 , 5 13 b in Quadrant II, find

41. Given sin a 苷

a. sin共a b兲

b. cos共a b兲

c. tan共a b兲

In Exercises 49 to 74, verify the identity.

冉冊

49. cos

p u 苷 sin u 2

50. cos共u p兲苷 cos u

冉冊

8 7 , a in Quadrant IV, and cos b 苷 , b 25 17 in Quadrant IV, find

51. sin u

a. sin共a b兲

52. sin共u p兲苷 sin u

42. Given sin a 苷

b. cos共a b兲

c. tan共a b兲

3 15 , a in Quadrant I, and sin b 苷 , b in 17 5 Quadrant III, find

43. Given cos a 苷

a. sin共a b兲

b. cos共a b兲

c. tan共a b兲

12 7 , a in Quadrant II, and sin b 苷 , b 25 13 in Quadrant IV, find

44. Given cos a 苷

a. sin共a b兲

b. cos共a b兲

a. sin共a b兲

b. cos共a b兲

c. tan共a b兲

24 8 , a in Quadrant IV, and sin b 苷 , b 17 25 in Quadrant III, find

46. Given cos a 苷

a. sin共a b兲

b. cos共a b兲

冉冊

c. tan共a b兲

苷 cos u

53. tan u

p 4

54. tan 2u 苷

2 tan u 1 tan2 u

苷

tan u 1 1 tan u

冉冊冉冊冉冊

55. cos

3p u 苷 sin u 2

56. sin

3p u 苷 cos u 2

57. cot

p u 苷 tan u 2

c. tan共a b兲

5 3 45. Given cos a 苷 , a in Quadrant III, and sin b 苷 , b 5 13 in Quadrant I, find

p 2

58. cot共p u兲苷 cot u 59. csc共p u兲苷 csc u

冉冊

60. sec

p u 苷 csc u 2

61. sin 6x cos 2x cos 6x sin 2x 苷 2 sin 2x cos 2x 62. cos 5x cos 3x sin 5x sin 3x 苷 cos2 x sin2 x

5 3 , a in Quadrant I, and tan b 苷 , b in 5 12 Quadrant III, find

47. Given sin a 苷

a. sin共a b兲

b. cos共a b兲

c. tan共a b兲

63. cos共a b兲 cos共a b兲苷 2 cos a cos b 64. cos共a b兲 cos共a b兲苷 2 sin a sin b 65. sin共a b兲 sin共a b兲苷 2 sin a cos b

7 15 48. Given tan a 苷 , a in Quadrant I, and tan b 苷 , b in 8 24 Quadrant IV, find a. sin共a b兲

b. cos共a b兲

c. tan共a b兲

66. sin共a b兲 sin共a b兲苷 2 cos a sin b 67.

cot a tan b cos共a b兲苷 sin共a b兲 1 cot a tan b

3.2

Sum, Difference, and Cofunction Identities

235

68.

sin共a b兲 1 cot a tan b 苷 sin共a b兲 1 cot a tan b

In Exercises 75 to 80, write the given expression as a function that involves only sin , cos , or tan . (In Exercises 78, 79, and 80, assume k is an integer.)

69.

sin h 共cos h 1兲 sin共x h兲 sin x 苷 cos x sin x h h h

75. cos共u 3p兲

76. sin共u 2p兲

77. tan共u p兲

78. cos关u 共2k 1兲p兴

79. sin共u 2kp兲

80. sin共u kp兲

共cos h 1兲 sin h cos共x h兲 cos x 苷 cos x sin x 70. h h h

冉冉

冊冊

71. sin

p a b 苷 cos a cos b sin a sin b 2

72. cos

p a b 苷共sin a cos b cos a sin b兲 2

73. sin 3x 苷 3 sin x 4 sin3 x 74. cos 3x 苷 4 cos x 3 cos x 3

In Exercises 81 to 84, compare the graphs of each side of the equation to predict whether the equation is an identity.

冉冊

81. sin

p x 苷 cos x 2

82. cos共x p兲苷 cos x

83. sin 7x cos 2x cos 7x sin 2x 苷 sin 5x 84. sin 3x 苷 3 sin x 4 sin3 x

Connecting Concepts In Exercises 85 to 89, verify the identity.

85. sin共x y兲 sin共x y兲苷 sin2 x cos2 y cos2 x sin2 y

swimming upward at angle b and then gliding down at angle a to the energy it expends swimming horizontally is given by

86. sin共x y z兲苷 sin x cos y cos z cos x sin y cos z cos x cos y sin z sin x sin y sin z 87. cos共x y z兲苷 cos x cos y cos z sin x sin y cos z sin x cos y sin z cos x sin y sin z 88.

sin共x y兲苷 cot x cot y sin x sin y

89.

cos共x y兲苷 cot y tan x cos x sin y

90. MODEL RESISTANCE The drag (resistance) on a fish when it is swimming is two to three times the drag when it is gliding. To compensate for this, some fish swim in a sawtooth pattern, as shown in the accompanying figure. The ratio of the amount of energy the fish expends when

ER 苷

k sin a sin b k sin共a b兲

where k is a value such that 2 k 3, and k depends on the assumptions we make about the amount of drag experienced by the fish. Find ER for k 苷 2, a 苷 10 , and b 苷 20 .

α β

236

Chapter 3

Trigonometric Identities and Equations

Projects 1. INTERSECTING LINES In the figure shown at the right, two nonvertical lines intersect in a plane. The slope of line l1 is m1 and the slope of line l2 is m2 .

y l2 l1

a. Show that the tangent of the smallest positive angle between l1 and l2 is given by tan 苷

m2 m1 1 m1 m 2

γ

α

β

b. Two nonvertical lines intersect at the point 共1, 5兲. The measure of the smallest positive angle between the lines is 苷 60 . The first line is given by y 苷 0.5x 4.5. What is the equation (in slope-intercept form) of the second line?

Section 3.3 ■ ■

■

Double-Angle Identities Power-Reducing Identities Half-Angle Identities

x

Double- and Half-Angle Identities P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A16.

PS1. Use the identity for sin共兲 to rewrite sin 2. [3.2] PS2. Use the identity for cos共a b兲 to rewrite cos 2. [3.2] PS3. Use the identity for tan共a b兲 to rewrite tan 2. [3.2] PS4. Compare tan

sin and for 苷 60 , 苷 90 , and 苷 120 . [2.2] 2 1 cos

PS5. Verify that sin 2 苷 2 sin is not an identity. [2.2] PS6. Verify that cos

Integrating Technology One way of showing that sin 2x 苷 2 sin x is by graphing y 苷 sin 2x and y 苷 2 sin x and observing that the graphs are not the same.

a 1 苷 cos a is not an identity. [2.2] 2 2

Double-Angle Identities By using the sum identities, we can derive identities for f共2a兲, where f is a trigonometric function. These are called the double-angle identities. To find the sine of a double angle, substitute for in the identity for sin共兲. sin 共a b兲苷 sin a cos b cos a sin b sin 共兲苷 sin cos cos sin sin 2 苷 2 sin cos

• Let .

3.3

Double- and Half-Angle Identities

237

A double-angle identity for cosine is derived in a similar manner. cos共兲苷 cos cos sin sin cos共兲苷 cos cos sin sin cos 2 苷 cos2 sin2

• Let .

There are two alternative forms of the double-angle identity for cos 2. Using cos2 苷 1 sin2 , we can rewrite the identity for cos 2 as follows: cos 2 苷 cos2 sin2 cos 2 苷共1 sin2 兲 sin2 cos 2 苷 1 2 sin2

• cos2 1 sin2

We also can rewrite cos 2 as cos 2 苷 cos2 sin2 cos 2 苷 cos2 共1 cos2 兲 cos 2 苷 2 cos2 1

• sin2 1 cos2

The double-angle identity for the tangent function is derived from the identity for tan共a b兲 with 苷 . tan a tan b 1 tan a tan b tan a tan a tan共a a兲苷 1 tan a tan a 2 tan tan 2 苷 1 tan2 tan共a b兲苷

• Let .

Here is a summary of the double-angle identities. Double-Angle Identities sin 2 苷 2 sin cos cos 2 苷 cos2 sin2 苷 1 2 sin2 苷 2 cos2 1 tan 2 苷

2 tan 1 tan2

The double-angle identities are often used to write a trigonometric expression in terms of a single trigonometric function. EXAMPLE 1

»

Simplify a Trigonometric Expression

Write 4 sin 5 cos 5 in terms of a single trigonometric function.

Solution 4 sin 5u cos 5u 苷 2共2 sin 5u cos 5u兲苷 2共sin 10u兲苷 2 sin 10u

»

Try Exercise 2, page 243

• Use 2 sin cos sin 2, with 5 .

238

Chapter 3

Trigonometric Identities and Equations

QUESTION

Does sin cos 苷

1 sin 2? 2

In Example 2, we use given information concerning the angle to find the exact value of sin 2.

EXAMPLE 2

If sin a 苷

»

Evaluate a Trigonometric Function

4 and 0 90 , find the exact value of sin 2. 5

Solution Use the identity sin 2 苷 2 sin cos . Find cos by substituting for sin in sin2 cos2 苷 1 and solving for cos . cos 苷兹1 sin2 苷

冑冉冊 1

4 5

2

苷

3 5

• cos w 0 if is in Quadrant I.

Substitute the values of sin and cos in the double-angle formula for sin 2. 4 3 24 sin 2a 苷 2 sin a cos a 苷 2 苷 5 5 25

冉冊冉冊

»

Try Exercise 10, page 243

EXAMPLE 3

»

Verify an Identity

Verify the identity csc 2 苷

1 共tan cot 兲. 2

Solution Work on the right-hand side of the equation. 1 1 共tan a cot a兲苷 2 2

冉冉

冊冊

sin a cos a cos a sin a

sin2 a cos2 a cos a sin a

苷

1 2

苷

1 1 苷苷 csc 2a 2 cos a sin a sin 2a

»

Try Exercise 50, page 244

ANSWER

Yes. sin cos 苷

2 sin cos sin 2 1 苷苷 sin 2. 2 2 2

3.3

Double- and Half-Angle Identities

239

Power-Reducing Identities The double-angle identities can be used to derive the following power-reducing identities. These identities can be used to write trigonometric expressions involving even powers of sine, cosine, and tangent in terms of the first power of a cosine function.

兰

Calculus Connection

Power-Reducing Identities sin2 a 苷

1 cos 2a 2

cos2 a 苷

1 cos 2a 2

tan2 a 苷

1 cos 2a 1 cos 2a

The first power-reducing identity is derived by solving the double-angle identity cos 2a 苷 1 2 sin2 a for sin2 a. The second identity is derived by solving the double-angle identity cos 2a 苷 2 cos2 a 1 for cos2 a. The identity for tan2 a can be derived by using the ratio identity, as shown below. 1 cos 2a 2 sin a 2 1 cos 2a tan2 a 苷苷苷 2 cos a 1 cos 2a 1 cos 2a 2

EXAMPLE 4

»

Use Power-Reducing Identities

Write sin4 a in terms of the first power of one or more cosine functions.

Solution sin4 a 苷 (sin2 a)2

»

苷

冉

冊

苷

1 (1 2 cos 2a cos2 2a) 4

苷

1 1 cos 4a 1 2 cos 2a 4 2

苷

1 4

苷

1 (3 4 cos 2a cos 4a) 8

1 cos 2 a 2

2

冉冉

• Power-reducing identity

• Square.

冊冊

2 4 cos 2a 1 cos 4a 2

Try Exercise 22, page 243

• Power-reducing identity

• Simplify.

240

Chapter 3

Trigonometric Identities and Equations

Half-Angle Identities The following identities, called half-angle identities, can be derived from the power-reducing identities by replacing with and taking the square root of 2 a each side. Two additional identities are given for tan . 2 Half-Angle Identities

冑冑冑

sin

a 苷

2

1 cos a 2

cos

a 苷

2

1 cos a 2

tan

a 苷

2

1 cos a sin a 1 cos a 苷苷 1 cos a 1 cos a sin a

The choice of the plus or minus sign depends on the quadrant in which

a lies. 2

In Example 5, we use a half-angle identity to find the exact value of a trigonometric function.

EXAMPLE 5

»

Evaluate a Trigonometric Function

Find the exact value of cos 105 .

Solution 1 (210 ), we can find cos 105 by using the half-angle 2 a a identity for cos with a 苷 210 . The angle 苷 105 lies in Quadrant II and 2 2 the cosine function is negative in Quadrant II. Thus cos 105 0, and we must a 1 cos a select the minus sign that precedes the radical in cos 苷

to 2 2 produce the correct result. Because 105 苷

冑

冑

cos 105 苷

1 cos 210 2

冑冑

冉冊

1

苷

苷

兹3 2

2 2 兹3 2 2 2

with 210 2 and a minus sign in front of the radical.

• Use the formula for cos

• cos 210°

兹3 2

• Simplify the numerator of the radicand.

3.3

冑冉冑

2 兹3 2

苷

冊

1 2

241

• Definition of division

2 兹3 4

苷

苷

Double- and Half-Angle Identities

• Simplify.

兹2 兹3 2

»

Try Exercise 28, page 243

In Example 6, we use given information concerning an angle to find the exact values of the cosine and tangent of . 2 EXAMPLE 6

If sin a 苷 cos

a.

a 2

»

Evaluate Trigonometric Functions

3 and 180 a 270 , find the exact value of 5 b.

tan

a 2

Solution To apply the half-angle identities, we need to find cos a. We can use the identity cos2 a 苷 1 sin2 a to find cos a, but first we need to determine the sign of cos a. Because 180 a 270 , we know that cos a 0. Thus

冉冊

冑

冑

a 苷 2

cos

3 5

冑

2

16 16 4 and cos a 苷苷 25 25 5 a 1 Multiply each part of 180 a 270 by to obtain 90 135 . 2 2 a a a Therefore, lies in Quadrant II. Thus cos 0 and tan 0. 2 2 2 a a. In the following work, we use the half-angle identity for cos with a 2 minus sign in front of the radical. cos2 a 苷 1 sin2 a 苷 1

1 cos a 苷 2

苷

冉冊冑

1 2

4 5

苷

We could use the half-angle identity tan

b.

the identity tan

tan

»

冑

1 1 苷 5 2

冑

a 苷

2

1 兹10 苷 10 10

1 cos a ; however, 1 cos a

a sin a 苷 is simpler and easier to evaluate. 2 1 cos a

a sin a 苷苷 2 1 cos a

3 5

冉冊

4 1 5

Try Exercise 38, page 243

冉冊

苷

3 5

冉冊

1 3 苷 5 5

5 苷 3

242

Chapter 3

Trigonometric Identities and Equations The power-reducing identities and half-angle identities can be used to verify other identities.

EXAMPLE 7

»

Verify an Identity

Verify the identity 2 csc x cos2

x sin x 苷 . 2 1 cos x

Solution Work on the left side of the identity. 2 csc x cos2

冉

冊

x 1 cos x 苷 2 csc x 2 2 1 cos x 苷 sin x 1 cos x 1 cos x 苷 sin x 1 cos x 1 cos2 x sin x共1 cos x兲 sin2 x 苷 sin x共1 cos x兲 sin x 苷 1 cos x

• cos2

x 1 cos x 2 2

• csc x

1 sin x

• Multiply the numerator and denominator by the conjugate of the numerator.

苷

• 1 cos2 x sin2 x • Simplify.

»

Try Exercise 68, page 244

Topics for Discussion 1. True or false: If sin 苷 sin , then 苷 . Why? 2. Because tan

sin 苷 2 1 cos

and

tan

1 cos 苷 2 sin

are both identities, it follows that sin 1 cos 苷 1 cos sin is also an identity. Do you agree? Explain. 3. Is sin 10x 苷 2 sin 5x cos 5x an identity? Explain. 4. Is sin

苷 cos an identity? Explain. 2 2

3.3

243

Double- and Half-Angle Identities

Exercise Set 3.3 In Exercises 1 to 8, write each trigonometric expression in terms of a single trigonometric function.

21. cos4 x

22. sin2 x cos4 x 24. sin6 x

1. 2 sin 2 cos 2

2. 2 sin 3 cos 3

23. sin4 x cos2 x

3. 1 2 sin2 5

4. 2 cos2 2 1

In Exercises 25 to 36, use the half-angle identities to find the exact value of each trigonometric expression.

5. cos2 3 sin2 3

6. cos2 6 sin2 6

7.

2 tan 3 1 tan2 3

8.

2 tan 4 1 tan2 4

In Exercises 9 to 18, find the exact values of sin 2, cos 2, and tan 2 given the following information.

9. cos a 苷

10. cos a 苷

4 5

24 25

8 11. sin a 苷 17 9 12. sin a 苷 41 24 13. tan a 苷 7 4 14. tan a 苷 3 15 15. sin a 苷 17 3 16. sin a 苷 5 40 17. cos a 苷 41 4 18. cos a 苷 5

25. sin 75°

26. cos 105°

27. tan 67.5°

28. cos 165°

29. cos 157.5°

30. sin 112.5°

31. sin 22.5°

32. cos 67.5°

33. sin

7 8

36. sin

3 8

34. cos

5 8

35. cos

5 12

90 a 180 In Exercises 37 to 44, find the exact values of the sine, co-

270 a 360

90 a 180

180 a 270

270 a 360

0 a 90

0 a 90

180 a 270

270 a 360

270 a 360

sine, and tangent of

37. sin 苷

5 13

38. sin 苷

given the following information. 2

90 a 180

7 25

180 a 270

8 17

180 a 270

39. cos 苷

40. cos 苷

12 13

0 a 90

41. tan 苷

4 3

0 a 90

42. tan 苷

43. cos 苷

8 15

24 25

44. sin 苷

9 41

90 a 180

270 a 360

270 a 360

In Exercises 45 to 90, verify the given identity. In Exercises 19 to 24, use the power-reducing identities to write each trigonometric expression in terms of the first power of one or more cosine functions.

45. sin 3x cos 3x 苷

19. 6 cos2 x

46. cos 8x 苷 cos2 4x sin2 4x

20. sin4 x cos4 x

1 sin 6x 2

244

Chapter 3

Trigonometric Identities and Equations x x cos 苷 sin x 2 2

47. sin2 x cos 2x 苷 cos2 x

48.

cos 2x 苷 cot 2 x 1 sin2 x

71. 2 sin

49.

1 cos 2x 苷 cot x sin 2x

50.

1 1 苷 csc2 x 1 cos 2x 2

73.

51.

sin 2x 苷 2 tan x 1 sin2 x

74. tan2

x sec x 1 苷 2 sec x 1

52.

cos2 x sin2 x 苷 cot 2x 2 sin x cos x

75. sin2

1 x sec x 苷共sec x 1兲 2 2

cos 2x cos2 x

76. cos2

1 x sec x 苷共sec x 1兲 2 2

2 sin x cos x cos2 x sin2 x

77. cos2

x x cos x 苷 sin2 2 2

78. sin2

x x cos x 苷 cos2 2 2

79. sin2

x x cos2 苷 cos x 2 2

80. cos2

x x 1 sin2 苷 csc x sin 2x 2 2 2

53. 1 tan2 x 苷

54. tan 2x 苷

55. sin 2x tan x 苷 tan x cos 2x

冉

56. sin 2x cot x 苷 cot x cos 2x 57. cos4 x sin4 x 苷 cos 2x 58. sin 4x 苷 4 sin x cos3 x 4 cos x sin3 x

x x sin 2 2

cos

59. cos2 x 2 sin2 x cos2 x sin2 x 2 sin4 x 苷 cos2 2x 60. 2 cos4 x cos2 x 2 sin2 x cos2 x sin2 x 苷 cos2 2x 61. cos 4x 苷 1 8 cos2 x 8 cos4 x

83. tan 2x 苷

冉冉

65. sin3 x cos3 x 苷共sin x cos x兲 1

1 sin 2x 2

66. cos3 x sin3 x 苷共cos x sin x兲 1

1 sin 2x 2

67. sin2

x sec x 1 苷 2 2 sec x

68. cos2

69. tan

x 苷 csc x cot x 2

70. tan

冊冊

x sec x 1 苷 2 2 sec x x tan x 苷 2 sec x 1

苷 1 sin x

cos 2x 苷 csc2 x 2 sin2 x

62. sin 4x 苷 4 sin x cos x 8 cos x sin3 x

64. sin 3x sin x 苷 4 sin x 4 sin3 x

2

x x sin2 苷 cos x 2 2

81. sin 2x cos x 苷共cos x兲共2 sin x 1兲 82.

63. cos 3x cos x 苷 4 cos3 x 4 cos x

冊

72. cos2

84.

2 cos 2x 苷 cot x tan x sin 2x

85. 2 tan

86.

2 cot x tan x

sin2 x 1 cos2 x x 苷 2 共sin x兲共1 cos x兲

x 1 csc2 苷 csc2 x cot x csc x 2 2

87. csc 2x 苷

89. cos

1 csc x sec x 2

x x 苷 1 2 sin2 5 10

88. sec 2x 苷

90. sec2

sec2 x 2 sec2 x

x 2 苷 2 1 cos x

3.3 91. MACH NUMBERS Ernst Mach (1838 – 1916) was an Austrian physicist who made a study of the motion of objects at high speeds. Today we often state the speed of aircraft in terms of a Mach number. A Mach number is the speed of an object divided by the speed of sound. For example, a plane flying at the speed of sound is said to have a speed M of Mach 1. Mach 2 is twice the speed of sound. An airplane that travels faster than the speed of sound creates a sonic boom. This sonic boom emanates from the airplane in the shape of a cone. Sonic boom cone

Double- and Half-Angle Identities

245

The following equation shows the relationship between the measure of the cone’s vertex angle and the Mach speed M of an aircraft that is flying faster than the speed of sound. M sin

苷1 2

, determine the Mach speed M of the airplane. 4 State your answer as an exact value and as a decimal accurate to the nearest hundredth.

a. If 苷

b. Solve M sin

a 苷 1 for . 2

c. Does the vertex angle increase or decrease as the Mach number M increases? Hyperbola

Connecting Concepts In Exercises 92 to 95, compare the graphs of each side of the equation to predict whether the equation is an identity.

92. sin2 x cos 2x 苷 cos2 x

94. sin

95.

冉

93.

sin 2x 苷 2 tan x 1 sin2 x

x x cos 苷 sin x 2 2

cos

x x sin 2 2

冊

2

In Exercises 96 to 98, verify the identity.

96.

1 sin3 x cos3 x 苷 1 sin 2x sin x cos x 2

97. cos4 x 苷

98.

1 1 3 cos 4x cos 2x 8 2 8

sin x sin 2x x 苷 tan cos x cos 2x 2

苷 1 sin x

Projects VISUAL INSIGHT Explain how the figure at the right can be used to verify the half-angle identity tan

sin 苷 2 1 cos

Unit circle

sin θ

1.

1 θ

cos θ

θ 2 1

246

Chapter 3

Trigonometric Identities and Equations

Section 3.4 ■

■

■

The Product-to-Sum Identities The Sum-to-Product Identities Functions of the Form f(x) a sin x b cos x

Identities Involving the Sum of Trigonometric Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A17.

PS1. Use sum and difference identities to rewrite

1 关sin共兲 sin共兲兴. [3.2] 2

PS2. Use sum and difference identities to rewrite

1 关cos共兲 cos共兲兴. [3.2] 2

PS3. Compare sin x sin y and 2 cos

xy xy p sin for x 苷 p and y 苷 . [2.4] 2 2 6

冉冊

PS4. Use a sum identity to rewrite 兹2 sin x

. [3.2] 4

PS5. Find a real number x and a real number y to verify that sin x sin y 苷 sin共x y兲 is not an identity. [2.4] PS6. Evaluate 兹a2 b2 for a 苷 1 and b 苷兹3 .

The Product-to-Sum Identities Some applications require that a product of trigonometric functions be written as a sum or difference of these functions. Other applications require that the sum or difference of trigonometric functions be represented as a product of these functions. The product-to-sum identities are particularly useful in these types of problems.

Product-to-Sum Identities sin a cos b 苷

1 关sin共a b兲 sin共a b兲兴 2

cos a sin b 苷

1 关sin共a b兲 sin共a b兲兴 2

cos a cos b 苷

1 关cos共a b兲 cos共a b兲兴 2

sin a sin b 苷

1 关cos共a b兲 cos共a b兲兴 2

3.4

Identities Involving the Sum of Trigonometric Functions

247

The product-to-sum identities can be derived by using the sum or difference identities. Adding the identities for sin共a b兲 and sin共a b兲, we have sin共a b兲苷 sin a cos b cos a sin b sin共a b兲苷 sin a cos b cos a sin b sin共a b兲 sin共a b兲苷 2 sin a cos b

• Add the identities.

Solving for sin a cos b, we obtain the first product-to-sum identity: sin a cos b 苷

1 关sin共a b兲 sin共a b兲兴 2

The identity for cos a sin b is obtained when sin共a b兲 is subtracted from sin共a b兲. The result is cos a sin b 苷

1 关sin共a b兲 sin共a b兲兴 2

In like manner, the identities for cos共a b兲 and cos共a b兲 are used to derive the identities for cos a cos b and sin a sin b. The product-to-sum identities can be used to verify some identities.

EXAMPLE 1

»

Verify an Identity

Verify the identity cos 2x sin 5x 苷

1 共sin 7x sin 3x兲. 2

Solution cos 2x sin 5x 苷

1 关sin共2x 5x兲 sin共2x 5x兲兴 2

1 关sin 7x sin共3x兲兴 2 1 苷共sin 7x sin 3x兲 2

苷

• Use the productto-sum identity for cos sin .

• sin(3x) sin 3x

»

Try Exercise 36, page 252

The Sum-to-Product Identities The sum-to-product identities can be derived from the product-to-sum identities.

248

Chapter 3

Trigonometric Identities and Equations

Sum-to-Product Identities sin x sin y 苷 2 sin

xy xy cos 2 2

cos x cos y 苷 2 cos

xy xy cos 2 2

sin x sin y 苷 2 cos

xy xy sin 2 2

cos x cos y 苷 2 sin

xy xy sin 2 2

To derive the sum-to-product identity for sin x sin y, we first let x 苷 a b and y 苷 a b. Then x y 苷 a b a b and x y 苷 a b 共a b兲 x y 苷 2a x y 苷 2b xy xy a苷 b苷 2 2 Substituting these expressions for a and b into the product-to-sum identity 1 关sin共a b兲 sin共a b兲兴苷 sin a cos b 2 yields

冉

sin

冊冉

冊

xy xy xy xy xy xy sin 苷 2 sin cos 2 2 2 2 2 2

Simplifying the left side, we have the sum-to-product identity. sin x sin y 苷 2 sin

xy xy cos 2 2

The other three sum-to-product identities can be derived in a similar manner. The proofs of these identities are left as exercises.

EXAMPLE 2

»

Write the Difference of Trigonometric Expressions as a Product

Write sin 4u sin u as the product of two functions.

Solution sin 4u sin u 苷 2 cos

»

Try Exercise 22, page 251

4u u 4u u 5u 3u sin 苷 2 cos sin 2 2 2 2

3.4

249

Identities Involving the Sum of Trigonometric Functions QUESTION

Does cos 4u cos 2u 苷 2 cos 3u cos u?

EXAMPLE 3

»

Verify the identity

Verify a Sum-to-Product Identity

sin 6x sin 2x 苷 tan 4x cot 2x. sin 6x sin 2x

Solution 6x 2x 6x 2x cos sin 6x sin 2x 2 2 苷 sin 6x sin 2x 6x 2x 6x 2x 2 cos sin 2 2 sin 4x cos 2x 苷 cos 4x sin 2x 苷 tan 4x cot 2x 2 sin

y

»

Try Exercise 44, page 252

P(a, b)

Functions of the Form f (x) a sin x b cos x k

b

α

a

Figure 3.6

x

The function given by the equation f共x兲苷 a sin x b cos x can be written in the form f共x兲苷 k sin共x a兲. This form of the function is useful in graphing and engineering applications because the amplitude, period, and phase shift can be readily calculated. Let P共a, b兲 be a point on a coordinate plane, and let a represent an angle in standard position whose terminal side contains P. See Figure 3.6. To rewrite y 苷 a sin x b cos x, multiply and divide the expression a sin x b cos x by 兹a2 b2. 兹a2 b2 共a sin x b cos x兲兹a2 b2 a b 苷兹a2 b2 sin x cos x 兹a2 b2 兹a2 b2

a sin x b cos x 苷

冉

冊

(1)

From the definition of the sine and cosine of an angle in standard position, let k 苷兹a2 b2,

cos a 苷

a , 兹a b2 2

and

sin a 苷

b 兹a b2 2

Substituting these expressions into Equation (1) yields a sin x b cos x 苷 k共cos a sin x sin a cos x兲 Now, using the identity for the sine of the sum of two angles, we have a sin x b cos x 苷 k sin共x a兲 Thus a sin x b cos x 苷 k sin共x a兲, where k 苷兹a2 b2 and a is the angle b a for which sin a 苷 and cos a 苷 . 2 2 2 兹a b 兹a b2 ANSWER

冉

Yes. cos 4 cos 2 苷 2 cos

冊冉

冊

4 2 4 2 cos 苷 2 cos 3 cos . 2 2

250

Chapter 3

Trigonometric Identities and Equations Functions of the Form a sin x b cos x a sin x b cos x 苷 k sin共x a兲 where k 苷兹a2 b 2, sin a 苷

EXAMPLE 4

»

b a , and cos a 苷 2 2 兹a b 兹a b 2 2

Rewrite a sin x b cos x

Rewrite sin x cos x in the form k sin共x a兲.

Solution π y = 2 sin x + 4 y = sin x + cos x

(

y

y=

1

Comparing sin x cos x to a sin x b cos x, a 苷 1 and b 苷 1. Thus 1 1 p k 苷兹12 12 苷兹2, sin a 苷 , and cos a 苷 . Thus a 苷 . 4 兹2 兹2

)

冉冊

sin x cos x 苷 k sin共x a兲苷兹2 sin x

2 sin x

p 4

»

Try Exercise 62, page 252

π

x

2π

冉冊

–1

p The graphs of y 苷 sin x cos x and y 苷兹2 sin x 4 p of y 苷兹2 sin x shifted units to the left. See Figure 3.7. 4

Figure 3.7

EXAMPLE 5 y P(−1,

»

are both the graph

Graph a Function of the Form f(x) a sin x b cos x

Graph: f共x兲苷 sin x 兹3 cos x

3)

Solution First, we write f共x兲 as k sin共x a兲. Let a 苷 1 and b 苷兹3; then k 苷兹共1兲2 共兹3 兲2 苷 2. The amplitude is 2. The point P共 1, 兹3 兲 is in the second quadrant (see Figure 3.8). Let a be an angle in standard position with P on its terminal side. Let a be the reference angle for a. Then

k=2 b=

3

α'

α

a = −1

Figure 3.8

x

兹3 2 p a 苷 3

sin a 苷

a 苷 p a 苷 p

2p p 苷 3 3

3.4 y

Substituting 2 for k and

y = 2 sin x

251

Identities Involving the Sum of Trigonometric Functions 2p for a in y 苷 k sin共x a兲, we have 3

冉冊

2

y 苷 2 sin x π

2π

x

2p 3

2p The phase shift is . The graph of f共x兲苷 sin x 兹3 cos x is the graph 3 2p of y 苷 2 sin x shifted units to the left. See Figure 3.9. 3

–2 f(x) = − sin x + 3 cos x

»

Figure 3.9

Try Exercise 70, page 252

Topics for Discussion 1. The exact value of sin 75 cos 15 is

2 兹3 . Do you agree? Explain. 4

2. Do you agree with the following work? Explain. cos 195 cos 105 苷 cos共195 105 兲苷 cos 300 苷

冉冊

3. The graphs of y1 苷 sin x cos x and y2 苷兹2 sin x you agree? Explain.

p 4

1 2

are identical. Do

4. Explain how to find the amplitude of the graph of y 苷 a sin x b cos x.

Exercise Set 3.4 In Exercises 1 to 8, write each expression as the sum or difference of two functions.

1. 2 sin x cos 2x

2. 2 sin 4x sin 2x

3. cos 6x sin 2x

4. cos 3x cos 5x

5. 2 sin 5x cos 3x

6. 2 sin 2x cos 6x

7. sin x sin 5x

8. cos 3x sin x

11. cos 157.5 sin 22.5

13p p cos 12 12

14. sin

7p 11p sin 12 12

15. sin

p 7p cos 12 12

16. cos

7p 17p sin 12 12

In Exercises 17 to 32, write each expression as the product of two functions.

17. sin 4u sin 2u

18. cos 5u cos 3u

19. cos 3u cos u

20. sin 7u sin 3u

10. sin 105 cos 15

21. cos 6u cos 2u

22. cos 3u cos 5u

12. sin 195 cos 15

23. cos u cos 7u

24. sin 3u sin 7u

In Exercises 9 to 16, find the exact value of each expression. Do not use a calculator.

9. cos 75 cos 15

13. sin

252

Chapter 3

Trigonometric Identities and Equations

25. sin 5u sin 9u

26. cos 5u cos u

27. cos 2u cos u

28. sin 2u sin 6u

u 29. cos cos u 2

u 3u 30. sin sin 4 2

u u 31. sin sin 2 3

u 32. cos u cos 2

51. y 苷

1 兹3 sin x cos x 2 2

53. y 苷

1 1 sin x cos x 2 2

54. y 苷

1 兹3 sin x cos x 2 2

56. y 苷

兹2 兹2 sin x cos x 2 2

34. 2 sin a sin b 苷 cos共a b兲 cos共a b兲

57. y 苷 p sin x p cos x

35. 2 cos 3x sin x 苷 2 sin x cos x 8 cos x sin3 x

58. y 苷 0.4 sin x 0.4 cos x

36. sin 5x cos 3x 苷 sin 4x cos 4x sin x cos x 37. 2 cos 5x cos 7x 苷 cos2 6x sin2 6x 2 cos2 x 1 38. sin 3x cos x 苷 sin x cos x共3 4 sin2 x兲

In Exercises 59 to 66, write the given equation in the form y k sin (x ), where the measure of is in radians.

59. y 苷 sin x cos x 1 兹3 sin x cos x 2 2

39. sin 3x sin x 苷 2 sin x 4 sin3 x

61. y 苷

40. cos 5x cos 3x 苷 8 sin2 x共2 cos3 x cos x兲

63. y 苷 10 sin x 10 兹3 cos x

41. sin 2x sin 4x 苷 2 sin x cos x共4 cos2 x 1兲

64. y 苷 3 sin x 3 兹3 cos x

42. cos 3x cos x 苷 4 cos3 x 2 cos x

65. y 苷 5 sin x 5 cos x

43.

sin 3x sin x 苷 cot 2x cos 3x cos x

44.

cos 5x cos 3x 苷 tan x sin 5x sin 3x

62. y 苷 sin x 兹3 cos x

66. y 苷 3 sin x 3 cos x

67. y 苷 sin x 兹3 cos x

68. y 苷兹3 sin x cos x

69. y 苷 2 sin x 2 cos x

70. y 苷 sin x 兹3 cos x

71. y 苷兹3 sin x cos x

72. y 苷 sin x cos x

73. y 苷 5 sin x 5 兹3 cos x

cos 4x cos 2x 苷 tan 3x sin 2x sin 4x

74. y 苷兹2 sin x 兹2 cos x

47. sin共x y兲 cos共x y兲苷 sin x cos x sin y cos y 48. sin共x y兲 sin共x y兲苷 sin x sin y 2

75. y 苷 6 兹3 sin x 6 cos x

2

In Exercises 49 to 58, write the given equation in the form y k sin (x ), where the measure of is in degrees.

49. y 苷 sin x cos x

60. y 苷兹3 sin x cos x

In Exercises 67 to 76, graph one cycle of the function. Do not use a graphing calculator.

sin 5x sin 3x 45. 苷 2 cos x 4 sin x cos3 x 4 sin3 x cos x 46.

1 兹3 sin x cos x 2 2

55. y 苷 3 sin x 3 cos x

In Exercises 33 to 48, verify the identity.

33. 2 cos a cos b 苷 cos共a b兲 cos共a b兲

52. y 苷

50. y 苷兹3 sin x cos x

76. y 苷 5 兹2 sin x 5 兹2 cos x TONES ON A TOUCH-TONE PHONE In Exercises 77 and 78, use the following information about touch-tone phones. Every tone made on a touch-tone phone is produced by adding a pair of sounds. The following chart shows the sound fre-

3.4

Identities Involving the Sum of Trigonometric Functions

f 1 f2 . What is the frequency of the tone produced 2 when the 5 key is pressed?

quencies used for each key on the telephone keypad. For example, the sound emitted by pressing 3 on the keypad is produced by adding a 1477-hertz (cycles per second) sound to a 697-hertz sound. An equation that models this tone is p(t) sin(2 1477t) sin(2 697t)

78. a. Write an equation of the form p共t兲苷 sin共2p f1t兲 sin共2p f2t兲

where p is the pressure on the eardrum and t is the time in seconds.

ABC

DEF

2

3

GHI

JKL

MNO

4

5

6

PRS

TUV

WXY

7

8

9

697 Hz

1

770 Hz

852 Hz

941 Hz

* 1209 Hz

OPER

0

#

1336 Hz

1477 Hz

Source: Data in chart from http://www.howstuffworks.com/ telephone2.htm

that models the tone produced by pressing the 8 key on a touch-tone phone. b. Use a sum-to-product identity to write your equation from a. in the form p共t 兲苷 A sin共Bpt兲 sin共Cpt兲 c. What is the frequency of the tone produced when the 8 key is pressed? (Hint: See c. of Exercise 77.) In Exercises 79 to 84, compare the graphs of each side of the equation to predict whether the equation is an identity.

79. sin 3x sin x 苷 2 sin x 4 sin3 x 80.

77. a. Write an equation of the form p共t兲苷 sin共2p f 1t兲 sin共2p f2t兲 that models the tone produced by pressing the 5 key on a touch-tone phone. b. Use a sum-to-product identity to write your equation from a. in the form

1 sin 3x sin x 苷 cos 3x cos x tan 2x

冉冉冉冉

冊冊冊冊

81. 兹 3 sin x cos x 苷 2 sin x

5p 6

82. 兹 3 sin x cos x 苷 2 sin x

5p 6

p共t 兲苷 A sin共Bpt兲 sin共Cpt兲

83.

1 p 兹3 sin x cos x 苷 sin x 2 2 3

c. When a sound of frequency f 1 is combined with a sound of frequency f 2, the combined sound has a frequency of

84.

1 p 兹3 sin x cos x 苷 sin x 2 2 6

Connecting Concepts 85. Derive the sum-to-product identity cos x cos y 苷 2 cos

xy xy cos 2 2

In Exercises 89 to 94, verify the identity.

89. sin 2x sin 4x sin 6x 苷 4 sin 3x cos 2x cos x 90. sin 4x sin 2x sin 6x 苷 4 cos 3x sin 2x cos x

86. Derive the product-to-sum identity 1 关cos共x y兲 cos共x y兲兴 2

91.

cos 10x cos 8x 苷 cot x sin 10x sin 8x

87. If x y 苷 180 , show that sin x sin y 苷 2 sin x.

92.

sin 10x sin 2x 2 tan 3x 苷 cos 10x cos 2x 1 tan2 3x

sin x sin y 苷

88. If x y 苷 360 , show that cos x cos y 苷 2 cos x.

253

254

Chapter 3

Trigonometric Identities and Equations

93.

sin 2x sin 4x sin 6x 苷 tan 4x cos 2x cos 4x cos 6x

94.

sin 2x sin 6x 苷 cot 2x cos 6x cos 2x

95. Verify that cos2 x sin2 x 苷 cos 2x by using a product-tosum identity.

97. Verify that a sin x b cos x 苷 k cos共x a兲, where a k 苷兹a2 b 2 and tan a 苷 . b 98. Verify that a sin cx b cos cx 苷 k sin共cx a兲, where b k 苷兹a2 b 2 and tan a 苷 . a

96. Verify that 2 sin x cos x 苷 sin 2x by using a product-tosum identity.

Projects 1.

BEATS If two tuning forks that are close in frequency are struck at the same time, the sound we hear fluctuates between a loud tone and silence. These regular fluctuations are called beats. The loud periods occur when the sound waves reinforce (interfere constructively with) one another, and the silent periods occur when the waves interfere destructively with one another. If the frequencies of the tuning forks are 442 cycles per second and 440 cycles per second then the pressure produced on our eardrums by each tuning fork, respectively, is given by Y1 苷 sin共2p 442x兲 and

Y2 苷 sin共2p 440x兲

where x is the time in seconds. The combined pressure produced when both tuning forks are struck simultaneously is modeled by Y3 苷 Y1 Y2

a. Graph Y3. Use a viewing window with Xmin 苷 0, Xmax 苷 1, Ymin 苷 2.1, Ymax 苷 2.1. To produce a graph that better illustrates the beats produced when both tuning forks are struck simultaneously, graph Y3, Y4 苷 2 cos共2px兲, and Y5 苷 2 cos共2px兲 together in the same window. What is the relationship between the graph of Y3 and the graphs of Y4 and Y5? b. Use a sum-to-product identity to write Y3 as a product. c. The rate of the beats produced by two sounds with the same intensity is the absolute value of the difference between their frequencies. Consider two tuning forks that are struck at the same time with the same force and held on a sounding board. The tuning forks have frequencies of 564 and 568 cycles per second, respectively. How many beats will be heard each second? d. A piano tuner strikes a tuning fork and a key on a piano that is supposed to have the same frequency as the tuning fork. The piano tuner notices that the sound produced by the piano is lower than that produced by the tuning fork. The piano tuner also notes that the combined sound of the piano and the tuning fork has 2 beats per second. How much lower is the frequency of the piano than the frequency of the tuning fork?

3.5

Section 3.5 ■

■

■

Inverse Trigonometric Functions Composition of Trigonometric Functions and Their Inverses Graphs of Inverse Trigonometric Functions

255

Inverse Trigonometric Functions

Inverse Trigonometric Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A17.

PS1. What is a one-to-one function? [1.3] PS2. State the horizontal line test. [1.3] PS3. Find f 关 g共x兲兴 given that f共x兲苷 2x 4 and g共x兲苷

1 x 2. [1.5] 2

PS4. If f and f 1 are inverse functions, then determine f 关 f 1共x兲兴 for any x in the domain of f 1. [1.6] PS5. If f and f 1 are inverse functions, then explain how the graph of f 1 is related to the graph of f. [1.6] PS6. Use the horizontal line test to determine whether the graph of y 苷 sin x, where x is any real number, is a one-to-one function. [1.3/2.5]

Inverse Trigonometric Functions Because the graph of y 苷 sin x fails the horizontal line test, it is not the graph of a one-to-one function. Therefore, it does not have an inverse function. Figure 3.10 shows the graph of y 苷 sin x on the interval 2 x 2 and the graph of the inverse relation x 苷 sin y. Note that the graph of x 苷 sin y does not satisfy the vertical line test and therefore is not the graph of a function. If the domain of the function y 苷 sin x is restricted to x , the graph 2 2 of y 苷 sin x satisfies the horizontal line test and therefore the function has an inverse function. The graphs of y 苷 sin x for x and its inverse are shown in 2 2 Figure 3.11 on the next page. y

y 2π

TO REVIEW

Inverse Functions See section 1.6.

π

1 2π −2π

−π

π

x

−2

−1

−1

1 −π

−2π

y 苷 sin x

x 苷 sin y

Figure 3.10

2

x

256

Chapter 3

Trigonometric Identities and Equations

y

To find the inverse of the function defined by y 苷 sin x, with interchange x and y. Then solve for y.

1

−

π 2

x , 2 2

π 2

−1

y 苷 sin x:

x

x 2 2

y 苷 sin x

•

x 苷 sin y y苷?

• Solve for y.

• Interchange x and y.

Unfortunately, there is no algebraic solution for y. Thus we establish new notation and write

x 2 2

y 苷 sin1 x

y

which is read “y is the inverse sine of x.” Some textbooks use the notation arcsin x instead of sin1 x.

π 2

−1

1

Definition of sin1 x

x

y 苷 sin1 x

π − 2

where 1 x 1 and

y 苷 sin1 x: 1 x 1

if and only if x 苷 sin y

y . 2 2

Figure 3.11

take note 1

The 1 in sin x is not an exponent. The 1 is used to denote the inverse function. To use 1 as an exponent for a sine function, enclose the function in parentheses. 共sin x兲1 苷 sin1 x 苷

1 苷 csc x sin x 1 sin x

冉冊

It is convenient to think of the value of an inverse trigonometric function as 1 an angle. For instance, if y 苷 sin1 , then y is the angle in the interval 2 1 whose sine is . Thus y 苷 , . 2 2 2 6 Because the graph of y 苷 cos x fails the horizontal line test, it is not the graph of a one-to-one function. Therefore, it does not have an inverse function. Figure 3.12 shows the graph of y 苷 cos x on the interval 2 x 2 and the graph of the inverse relation x 苷 cos y. Note that the graph of x 苷 cos y does not satisfy the vertical line test and therefore is not the graph of a function. If the domain of y 苷 cos x is restricted to 0 x , the graph of y 苷 cos x satisfies the horizontal line test and therefore is the graph of a one-to-one function. The graphs of y 苷 cos x for 0 x and its inverse are shown in

冋

册

Figure 3.13. y

y 2π

1

−2π

−π

π

π

2π

x

−1

−1

1 −π

−2π

y 苷 cos x

x 苷 cos y

Figure 3.12

x

3.5

Inverse Trigonometric Functions

257

To find the inverse of the function defined by y 苷 cos x, with 0 x , interchange x and y. Then solve for y.

y

y 苷 cos x x 苷 cos y y苷?

1 π 2

−1

π

x

y 苷 cos x: 0 x

•0x • Interchange x and y. • Solve for y.

As in the case for the inverse sine function, there is no algebraic solution for y. Thus the notation for the inverse cosine function becomes y 苷 cos1 x. We can write the following definition of the inverse cosine function. Definition of cos1 x

y

y 苷 cos1 x

π

if and only if x 苷 cos y

where 1 x 1 and 0 y . π 2

−1

1 −

π 2

y 苷 cos1 x: 1 x 1

Figure 3.13

x

Because the graphs of y 苷 tan x, y 苷 csc x, y 苷 sec x, and y 苷 cot x fail the horizontal line test, these functions are not one-to-one functions. Therefore, these functions do not have inverse functions. If the domains of all these functions are restricted in a certain way, however, the graphs satisfy the horizontal line test. Thus each of these functions has an inverse function over a restricted domain. Table 3.2 on page 258 shows the restricted function and the inverse function for tan x, csc x, sec x, and cot x. The choice of ranges for y 苷 sec1 x and y 苷 csc1 x is not universally ac 3 cepted. For example, some calculus texts use 0, , as the range of 2 2 y 苷 sec1 x. This definition has some advantages and some disadvantages that are explained in more advanced mathematics courses.

冋冊冋冊

EXAMPLE 1

»

Evaluate Inverse Functions

Find the exact value of each inverse function. y 苷 tan1

a.

兹3 3

b.

冉冊

y 苷 cos1

兹2 2

Solution 兹3 , y is the angle whose measure is in the interval 3 兹3 , , and tan y 苷 . Therefore, y 苷 . 2 2 3 6

Because y 苷 tan1

a.

冉

冊

冉冊

兹2 , y is the angle whose measure is in the 2 3 兹2 interval 关0, 兴, and cos y 苷 . Therefore, y 苷 . 2 4 Because y 苷 cos1

b.

»

Try Exercise 2, page 265

258

Chapter 3

Trigonometric Identities and Equations

Table 3.2

Domain

y tan1 x

x 2 2

x

y

Range

Asymptotes

y tan x

x苷

Graph −

,x苷 2 2

y

1

π 2

−1

π 2

x

−

y sec x Domain

Range

0 x , x 苷

2

x苷

x苷0

π 2 π 2

x

−1 −

π 2

x

1

y cot1 x

x 1 or x 1

0x

x

y

0y

x 苷 0, x 苷

y 苷 0, y 苷

y苷

2

2

y

π 2

1

Graph x

1

y cot x

2

π

y

−1

y π

π 2

π 2

y苷0

y

x

1

y ,y苷0 2 2

y sec1 x

y

–1

π 2

x 1 or x 1

y 1 or y 1

−

0 y , y 苷

y 1 or y 1

Asymptotes

−1

x ,x苷0 2 2

,y苷 2 2

y

π 2

y 2 2

y苷

y csc1 x

y csc x

–1

y π π 2

1 1

x –1

π x

π 2

–1

1

x

A calculator may not have keys for the inverse secant, cosecant, and cotangent functions. The following procedure shows an identity for the inverse cosecant function in terms of the inverse sine function. If we need to determine y, which is the angle whose cosecant is x, we can rewrite y 苷 csc1 x as follows: y 苷 csc1 x

• Domain: x 1 or x 1 Range:

y ,y0 2 2

csc y 苷 x

• Definition of inverse function

1 苷x sin y

• Substitute

1 for csc y. sin y

3.5 sin y 苷

Inverse Trigonometric Functions

1 x

259

• Solve for sin y.

y 苷 sin1

1 x

• Write using inverse notation.

csc1 x 苷 sin1

1 x

• Replace y with csc1 x.

Thus csc1 x is the same as sin1

1 . There is a similar identity for sec1 x. x

Identities for the Inverse Secant, Cosecant, and Cotangent Functions If x 1 or x 1, then csc1 x 苷 sin1

1 x

and

sec1 x 苷 cos1

1 x

If x is a real number, then cot 1 x 苷

tan1 x 2

Composition of Trigonometric Functions and Their Inverses

TO REVIEW

Composition of Functions See section 1.5.

Recall that a function f and its inverse f 1 have the property that f 关 f 1共x兲兴苷 x for all x in the domain of f 1 and that f 1关 f共x兲兴苷 x for all x in the domain of f. Applying this property to the functions sin x, cos x, and tan x and their inverse functions produces the following theorems. Composition of Trigonometric Functions and Their Inverses ■

If 1 x 1, then sin共sin1 x兲苷 x and cos共cos1 x兲苷 x.

■

If x is any real number, then tan共tan1 x兲苷 x.

■

If

■

If 0 x , then cos1共cos x兲苷 x.

■

If

x , then sin1共sin x兲苷 x. 2 2 x , then tan1共tan x兲苷 x. 2 2

In the next example we make use of some of the composition theorems to evaluate trigonometric expressions.

260

Chapter 3

Trigonometric Identities and Equations EXAMPLE 2

»

Evaluate the Composition of a Function and Its Inverse

Find the exact value of each composition of functions. a.

sin共sin1 0.357兲

b.

cos1共cos 3兲

c.

tan关tan1 共11.27兲兴

d.

sin共sin1 兲

e.

cos共cos1 0.277兲

f.

tan1 tan

冉冊 4 3

Solution a.

Because 0.357 is in the interval 关1, 1兴, sin共sin1 0.357兲苷 0.357.

b.

Because 3 is in the interval 关0, 兴, cos1共cos 3兲苷 3.

c.

Because 11.27 is a real number, tan关tan1 共11.27兲兴苷 11.27.

d.

Because is not in the domain of the inverse sine function, sin共sin1 兲 is undefined.

e.

Because 0.277 is in the interval 关1, 1兴, cos共cos1 0.277兲苷 0.277.

冉

冊

4 is not in the interval , ; however, the reference angle for 3 2 2 4 4 苷 is 苷 . Thus tan1 tan 苷 tan1 tan . Because 3 3 3 3 3 1 is in the interval , , tan tan 苷 . Hence 2 2 3 3 4 tan1 tan 苷 . 3 3

f.

冉冊

冉

冉冊冉冊冊冉冊

»

Try Exercise 28, page 265

QUESTION

Is tan1共tan x兲苷 x an identity?

It is often easy to evaluate a trigonometric expression by referring to a sketch of a right triangle that satisfies given conditions. In Example 3 we make use of this technique.

EXAMPLE 3

»

Evaluate a Trigonometric Expression

冉

Find the exact value of sin cos1

ANSWER

冊

2 . 5

No. tan1共tan x兲苷 x only if

x . 2 2

3.5

Inverse Trigonometric Functions

261

Solution

y

2 2 , which implies that cos 苷 . Because cos is positive, is a 5 5 first-quadrant angle. We draw a right triangle with base 2 and hypotenuse 5 so that we can view , as shown in Figure 3.14. The height of the triangle is 兹52 22 苷兹21. Our goal is to find sin , which by definition is opp 兹21 苷 . Thus hyp 5 Let 苷 cos1

5

52 − 22 =

21

θ

冉

x

2

Figure 3.14

sin cos1

2 5

冊

苷 sin共u兲苷

兹21 5

»

Try Exercise 50, page 265

In Example 4, we sketch two right triangles to evaluate the given expression.

EXAMPLE 4

»

Evaluate a Trigonometric Expression

冋

Find the exact value of sin sin1

冉冊册

3 5 cos1 5 13

Solution y

冉冊

3 3 5 . Thus sin a 苷 . Let 苷 cos1 , which implies 5 5 13 5 that cos b 苷 . Sketch angles and as shown in Figure 3.15. We wish to 13 evaluate Let 苷 sin1

5

.

3

冋

α

sin sin1

x

52 − 32 = 4

冉冊册

3 5 cos1 5 13

y

苷 sin共a b兲苷 sin cos cos sin

(1)

A close look at the triangles in Figure 3.15 shows that cos a 苷 13 2 − (5) 2 = 12

4 5

sin b 苷

and

12 13

13

Substituting in Equation (1) gives us our desired result. β' −5

Figure 3.15

冋

sin sin1

β

冉冊册

3 5 cos1 5 13

x

苷 sin cos cos sin 苷

冉冊冉冊冉冊冉冊 3 5

5 13

4 5

12 33 苷 13 65

»

Try Exercise 56, page 266

In Example 5 we make use of the identity cos共cos1 x兲苷 x, where 1 x 1, to solve an equation.

262

Chapter 3

Trigonometric Identities and Equations

EXAMPLE 5

Solve sin1

»

Solve an Inverse Trigonometric Equation

3 cos1 x 苷 . 5

Solution Solve for cos1 x and then take the cosine of both sides of the equation. sin1

3 cos1 x 苷 5 cos1 x 苷 sin1

冉

3 5

cos共cos1 x兲苷 cos sin1

5

3

3 5

冊

x 苷 cos共兲

α

• Let sin1

3 . Note 5

that is the angle whose

4

sine is

Figure 3.16

3 . (See Figure 3.16.) 5

苷 cos p cos a sin p sin a • Difference identity for cosine 苷共1兲 cos a 共0兲 sin a 苷 cos 4 4 • cos (See Figure 3.16.) 苷 5 5

»

Try Exercise 66, page 266

EXAMPLE 6

»

Verify a Trigonometric Identity That Involves Inverses

Verify the identity sin1 x cos1 x 苷 1

x

. 2

Solution Let 苷 sin1 x and 苷 cos1 x. These equations imply that sin 苷 x and cos 苷 x. From the right triangles in Figure 3.17,

α 1 − x2

cos 苷兹1 x2

sin 苷兹1 x2

and

Our goal is to show sin1 x cos1 x equals

1

1 − x2

β x

Figure 3.17

sin1 x cos1 x 苷苷 cos1关cos共兲兴

. 2

• Because 0 , we can apply

cos1[cos ( )].

3.5

Inverse Trigonometric Functions

苷 cos1关cos cos sin sin 兴

263

• Addition identity for cosine

苷 cos1关共兹1 x 2 兲共x兲共x兲共兹1 x 2 兲兴苷 cos1 0 p 苷 2

»

y

Try Exercise 72, page 266

π 2

−2

x

2

Graphs of Inverse Trigonometric Functions

−1

y = sin x −

π 2

y = sin−1 (x − 2)

Figure 3.18

The inverse trigonometric functions can be graphed by using the procedures of stretching, shrinking, and translation that were discussed earlier in the text. For instance, the graph of y 苷 sin1共x 2兲 is a horizontal shift 2 units to the right of the graph of y 苷 sin1 x, as shown in Figure 3.18.

EXAMPLE 7

»

Graph an Inverse Function

Graph: y 苷 cos1 x 1 y

Solution π

y = cos−1 x

Recall that the graph of y 苷 f共x兲 c is a vertical translation of the graph of f. Because c 苷 1, a positive number, the graph of y 苷 cos1 x 1 is the graph of y 苷 cos1 x shifted 1 unit up. See Figure 3.19.

y = cos−1 x + 1

»

π 2

Try Exercise 76, page 266

−1

1

x

Figure 3.19

Integrating Technology

2π

−3

3

− 2π

y 苷 3 sin1 0.5x

Figure 3.20

When you use a graphing utility to draw the graph of an inverse trigonometric function, use the properties of these functions to verify the correctness of your graph. For instance, the graph of y 苷 3 sin1 0.5x is shown in Figure 3.20. The domain of y 苷 sin1 x is 1 x 1. Therefore, the domain of y 苷 3 sin1 0.5x is 1 0.5x 1 or, multiplying the inequality by 2, 2 x 2. This is consistent with the graph in Figure 3.20. The range of y 苷 sin1 x is y . Thus the range of 2 2 3 3 y 苷 3 sin1 0.5x is y . This is also consistent with the graph. 2 2 Verifying some of the properties of y 苷 sin1 x serves as a check that you have correctly entered the equation for the graph.

264

Chapter 3

Trigonometric Identities and Equations EXAMPLE 8

»

Solve an Application

A camera is placed on a deck of a pool as shown in Figure 3.21. A diver is 18 feet above the camera lens. The extended length of the diver is 8 feet. Show that the angle u subtended at the lens by the diver is

a.

苷 tan1

Diver 8 ft

26 18 tan1 x x

b.

For what values of x will 苷 9 ?

c.

What value of x maximizes u?

Solution

β

From Figure 3.21 we see that 苷 tan1

b.

Use a graphing utility to graph 苷 tan1

α x

Figure 3.21

冉

冊

26 18 tan1 and x x

9 苷 radians . See Figure 3.22. Use the “intersect” command 20 20 to show that u is 9° for x ⬇ 12.22 feet and x ⬇ 38.29 feet. 苷

0.2

θ=

π 20

0.2

(12.22, 20π ) (38.29, π ) 20

θ, in radians

Camera

26 18 and 苷 tan1 . Because x x 26 18 苷 , we have 苷 tan1 tan1 . x x

a.

θ, in radians

18 ft

θ

26 18 θ = tan−1 x − tan−1 x

Maximum 0 X=21.633304 Y=.18283514 0 x, in feet

50

0 0

x, in feet

Figure 3.22

50

Figure 3.23

Use the “maximum” command to show that the maximum value of 26 18 苷 tan1 tan1 occurs when x ⬇ 21.63 feet. See Figure 3.23. x x

c.

»

Try Exercise 84, page 266

Topics for Discussion 1. Is the equation tan1 x 苷

1 tan x

true for all values of x, true for some values of x, or false for all values of x?

3.5

265

Inverse Trigonometric Functions

2. Are there real numbers x for which the following is true? Explain. sin共sin1 x兲苷 sin1共sin x兲 3. Explain how to find the value of sec1 3 by using a calculator. 4. Explain how you can determine the range of y 苷共2 cos1 x兲 1 using a. algebra

b. a graph

Exercise Set 3.5 In Exercises 1 to 18, find the exact radian value.

1. sin1 1

2. sin1

冉冊 1 2

4. cos1

7. cot 1

兹3 3

2 兹3 10. sec1 3

冉冊

16. cos1

冉冊

3. cos1

兹3 2

5. tan1共1兲

6. tan1 兹 3

8. cot1 1

9. sec1 2

11. csc1共兹 2 兲

兹3 2

17. tan1

12. csc1共2兲

冉冊

1 2

1 2

15. cos1

兹3 3

18. tan1 1

In Exercises 19 to 22, use a calculator to approximate each function accurate to four decimal places.

19. a. sin1(0.8422)

b. tan1(0.2385)

20. a. cos1(0.0356)

b. tan1(3.7555)

21. a. sec1(2.2500)

b. cot 1(3.4545)

22. a. csc1(1.3465)

b. cot 1(0.1274)

In Exercises 23 to 24, express p as a function of x.

24. θ 7

x

θ x

冊

1 2

27. tan共tan1 2兲

14. sin1

23.

冉

25. cos cos1

兹3 2

13. sin1

兹2 2

In Exercises 25 to 58, find the exact value of the given expression. If an exact value cannot be given, give the value to the nearest ten-thousandth.

5

冉冉

冊

29. sin tan1

3 4

31. tan sin1

兹2 2

冊

33. cos共sec1 2兲

冉冉冉冉冋冉冉冉

35. sin1 sin

6

37. cos1 sin

4

39. sin1 tan

3

41. tan1 sin

6

26. cos共cos1 2兲

冉冊冉冊冋冉冊册

28. tan tan1

1 2

30. cos sin1

5 13

32. sin cos1 34. sin1共sin 2兲

冊冊冊冊

冉冊册冊冊冊

43. sin1 cos

2 3

冉冉冉冉冋

36. sin1 sin

冊冊冊冊

5 6

38. cos1 cos

5 4

40. cos1 tan

2 3

42. cot 1 cos

2 3

冉冊册

44. cos1 tan

45. tan sin1

1 2

46. cot共csc1 2兲

47. sec sin1

1 4

48. csc cos1

3 4

7 25

50. tan cos1

3 5

49. cos sin1

兹3 2

冉冉

3

冊冊

266

Chapter 3

冉冉冉冉冉冉

51. cos 2 sin1 1

53. sin 2 sin

冊

冉

兹2 2 4 5

冊

Trigonometric Identities and Equations 52. tan 2 sin1 1

54. cos共2 tan

冊

In Exercises 75 to 82, use stretching, shrinking, and translation procedures to graph each equation.

兹3 2 1兲

冊冊冊冊

55. sin sin1

2 1 cos1 3 2

56. cos sin1

3 5 cos1 4 13

57. tan cos1

1 3 sin1 2 4

58. sec cos1

2 2 sin1 3 3

75. y 苷 sin1 x 2

76. y 苷 cos1共x 1兲

77. y 苷 sin1共x 1兲 2

78. y 苷 tan1共x 1兲 2

79. y 苷 2 cos1 x

80. y 苷 2 tan1 x

81. y 苷 tan1共x 1兲 2

82. y 苷 sin1共x 2兲 1

83. SATELLITE COVERAGE A communications satellite orbits Earth at an altitude of a miles. The beam coverage provided by the satellite is shown in the figure below.

s a

In Exercises 59 to 68, solve the equation for x algebraically.

59. sin1 x 苷 cos1

5 13

61. sin1 共x 1兲苷

2

冉

63. tan1 x

冊

兹2 苷 2 4

65. sin1

3 cos1 x 苷 5 4

67. sin1

2 兹2 cos1 x 苷 2 3

68. cos1 x sin1

60. tan1 x 苷 sin1

冉冊

24 25 苷

3

64. sin1共x 2兲苷

6

62. cos1 x

1 2

66. sin1 x cos1

4 苷 5 6

a. Find the distance s (length of arc MN) as a function of the altitude a of the satellite. Assume the radius of Earth is 3960 miles. b. What altitude does the satellite need to attain to provide coverage over a distance of s 苷 5500 miles? Round to the nearest 10 miles.

兹3 苷 2 2

84. VOLUME IN A WATER TANK The volume V of water (measured in cubic feet) in a horizontal cylindrical tank of radius 5 feet and length 12 feet is given by

冋

In Exercises 69 and 70, write each expression in terms of x.

69. tan共cos1 x兲

冉冊

V共x兲苷 12 25 cos1

70. sin共sec1 x兲

册

5x 共5 x兲兹10x x 2 5

12 ft

In Exercises 71 to 74, verify the identity. 5 ft

71. sin1 x sin1共x兲苷 0 72. cos1 x cos1共x兲苷 73. tan1 x tan1

x

Water

1 苷 ,x0 x 2 where x is the depth of the water in feet.

74. sec1

1 1 csc1 苷 x x 2

a.

Graph V over its domain 0 x 10.

3.5 b.

Write a sentence that explains why the graph of V increases more rapidly when x increases from 4.9 feet to 5 feet than it does when x increases from 0.1 foot to 0.2 foot.

267

Inverse Trigonometric Functions

86. Graph y 苷 cos共cos1 x兲 on 关1, 1兴. Graph y 苷 cos1 共cos x兲 on 关2, 2兴. In Exercises 87 to 92, use a graphing utility to graph each equation.

c. If x 苷 4 feet, find the volume (to the nearest 0.01 cubic foot) of the water in the tank.

87. y 苷 csc1 2x

88. y 苷 0.5 sec1

d. Find the depth x (to the nearest 0.01 foot) if there are 288 cubic feet of water in the tank.

89. y 苷 sec1共x 1兲

90. y 苷 sec1共x 兲

91. y 苷 2 tan1 2x

92. y 苷 tan1共x 1兲

85. Graph f 共x兲苷 cos1 x and g共x兲苷 sin1 兹1 x2 on the same coordinate axes. Does f 共x兲苷 g共x兲 on the interval 关1, 1兴?

x 2

Connecting Concepts In Exercises 97 to 100, solve for y in terms of x.

In Exercises 93 to 96, verify the identity. 1

93. cos共sin

x兲苷兹1 x 2

94. sec共sin1 x兲苷

95. tan共csc1 x兲苷

96. sin共cot 1 x兲苷

197. 5x 苷 tan1 3y

兹1 x 2 1 x2 兹x 2 1 ,x1 x2 1

99. x

苷 cos1共 y 3兲 3

100. x

苷 tan1共2y 1兲 2

兹x 2 1 x2 1

Projects 1. VISUAL INSIGHT

1

γ

β

α 1

1

1

Explain how the figure above can be used to verify each identity. a. tan1

1 1 tan1 苷 [Hint: Start by using an identity to find the value of tan共兲.] 3 2 4

b. 苷

98. 2x 苷

1 sin1 2y 2

268

Chapter 3

Section 3.6 ■

■

Solve Trigonometric Equations Model Sinusoidal Data

Trigonometric Identities and Equations

Trigonometric Equations P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A18.

PS1. Use the quadratic formula to solve 3x 2 5x 4 苷 0. [1.1] PS2. Use a Pythagorean identity to write sin2 x as a function involving cos2 x. [3.1] PS3. Evaluate

p 2kp for k 苷 1, 2, and 3. 2

PS4. Factor by grouping: x 2 PS5.

兹3 兹3 xx . 2 2

Graph the scatter plot for the following data. Use a viewing window with Xmin=0, Xmax=40, Ymin=0, and Ymax=100. [1.7] x

3

7

11

15

19

23

27

31

y

14

55

90

99

80

44

8

4

PS6. Solve 2x 2 2x 苷 0 by factoring. [1.1]

Solve Trigonometric Equations Consider the equation sin x 苷

1 . The graph of y 苷 sin x, along with the line 2

1 , is shown in Figure 3.24. The x values of the intersections of the two 2 1 graphs are the solutions of sin x 苷 . The solutions in the interval 0 x 2p 2 p 5p . are x 苷 and 6 6

y苷

y y=

1

−2π −1

π 6

5π 6

2π y = sin x

Figure 3.24

1 2

4π

x

3.6

269

Trigonometric Equations

If we remove the restriction 0 x 2p, there are many more solutions. Because the sine function is periodic with a period of 2p, other solutions are obtained by adding 2kp, k an integer, to either of the previous solutions. Thus the solutions 1 of sin x 苷 are 2 p x苷 2kp, k an integer 6 5p x苷 2kp, k an integer 6

QUESTION

How many solutions does the equation cos x 苷 interval 0 x 2p?

兹3 have on the 2

Algebraic methods and trigonometric identities are used frequently to find the solutions of trigonometric equations. Algebraic methods that are often employed include solving by factoring, solving by using the quadratic formula, and squaring each side of the equation.

EXAMPLE 1

»

Solve a Trigonometric Equation by Factoring

Solve 2 sin2 x cos x cos x 苷 0, where 0 x 2p.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

2 sin x cos x cos x 苷 0 cos x共2 sin2 x 1兲苷 0 cos x 苷 0 or 2 sin2 x 1 苷 0

The solutions are the x-coordinates of the x-intercepts of the graph of y 苷 2 sin2 x cos x cos x on the interval 关0, 2p兲.

2

x苷

p 3p , 2 2

sin2 x 苷

1 2

• Factor cos x from each term. • Use the Principle of Zero Products. • Solve each equation for x with 0 x 2.

兹2 sin x 苷

2 p 3p 5p 7p x苷 , , , 4 4 4 4

y 1

( 3π2 , 0) π

(

The solutions in the interval 0 x 2p are p p 3p 5p 3p 7p , , , , , and 4 2 4 4 2 4

( π2 , 0)

−1

( π4 , 0)

2π

)

3π ,0 4

( 5π4 , 0) (

x 7π ,0 4

)

y 苷 2 sin2 x cos x cos x

»

Try Exercise 14, page 278

ANSWER

Two

270

Chapter 3

Trigonometric Identities and Equations Squaring both sides of an equation may not produce an equivalent equation. Thus, when this method is used, the proposed solutions must be checked to eliminate any extraneous solutions.

EXAMPLE 2

»

Solve a Trigonometric Equation by Squaring Each Side of the Equation

Solve sin x cos x 苷 1, where 0 x 2p.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

sin x cos x 苷 1 sin x 苷 1 cos x sin2 x 苷共1 cos x兲2 sin2 x 苷 1 2 cos x cos2 x 1 cos2 x 苷 1 2 cos x cos2 x 2 2 cos x 2 cos x 苷 0 2 cos x共cos x 1兲苷 0 2 cos x 苷 0 or cos x 苷 1 p 3p x苷 , x苷0 2 2 A check will show that 0 and

• Solve for sin x.

• Square each side. • sin2 x 1 cos2 x • Factor.

• Solve each equation for x with 0 x 2.

p 3p are solutions but is not a solution. 2 2

The solutions are the x-coordinates of the points of intersection of y 苷 sin x cos x and y 苷 1 on the interval 关0, 2p兲. y

(0, 1)

y=1

( π2 , 1) π

2π

x

y = sin x + cos x

»

Try Exercise 52, page 278

EXAMPLE 3

»

Solve a Trigonometric Equation by Using the Quadratic Formula

Solve 3 cos2 x 5 cos x 4 苷 0, where 0 x 2p.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

The given equation is quadratic in form and cannot be factored easily. However, we can use the quadratic formula to solve for cos x.

The solutions are the x-coordinates of the x-intercepts of y 苷 3 cos2 x 5 cos x 4 on the interval 关0, 2p兲. See the following figure.

• a 3, b 5, c 4 3 cos2 x 5 cos x 4 苷 0 共5兲兹共5兲2 4共3兲共4兲 5 兹73 cos x 苷苷共2兲共3兲 6

The equation cos x 苷

5 兹73 does not have a solution because 6

5 兹73 2, and for any x the maximum value of cos x is 1. Thus 6

3.6 5 兹73 5 兹73 , and because is a negative number (about 6 6 5 兹73 0.59), the equation cos x 苷 will have two solutions on the 6 interval 关0, 2兲. Thus

y

cos x 苷

冉

冊

4 2 (2.2027, 0)

冉

(4.0805, 0) π

−2

5 兹73 ⬇ 2.2027 or 6 5 兹73 ⬇ 4.0805 x 苷 2 cos1 6 x 苷 cos1

271

Trigonometric Equations

2π

x

−4 −6

冊

y 苷 3 cos2 x 5 cos x 4

To the nearest 0.0001, the solutions on 关0, 2兲 are 2.2027 and 4.0805.

»

Try Exercise 56, page 278

When solving an equation that has multiple solutions, we must be sure we find all solutions of the equation for the given interval. For example, to find all 1 solutions of sin 2x 苷 , where 0 x 2, we first solve for 2x. 2 1 2 2x 苷 2k 6

sin 2x 苷

or

Solving for x, we have x 苷

2x 苷

5 2k 6

• k is an integer.

5 k or x 苷 k. Substituting integers 12 12

for k, we obtain k 苷 0: k 苷 1: k 苷 2:

5 or x 苷 12 12 13 17 x苷 or x 苷 12 12 25 29 x苷 or x 苷 12 12

Note that for k 2, x 2 and the solu1 tions to sin 2x 苷 are not in the inter2 val 0 x 2. Also if k 0, then x 0 and no solutions in the interval 0 x 2p are produced. Thus, for 0 x 2, the 5 13 17 solutions are , , , and . See 12 12 12 12 Figure 3.25.

y

x苷

( 12π , 12 ) y=

1

, 1 ( 13π 12 2 ) 17π 1 ( 12 , 2 ) , 1) ( 5π 12 2

1 2

x π

−1

y = sin 2x

Figure 3.25

2π

272

Chapter 3

EXAMPLE 4

»

Trigonometric Identities and Equations

Solve a Trigonometric Equation

Solve: sin 3x 苷 1

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

The equation sin 3x 苷 1 implies

The solutions are the x-coordinates of the points of intersection of y 苷 sin 3x and y 苷 1. The following figure shows eight of the points of intersection.

2k, k an integer 2 2k , k an integer x苷 6 3

3x 苷

• Divide each side by 3.

x苷

2k , where k is an integer 6 3

y=1

y

Because x is not restricted to a finite interval, the given equation has an infinite number of solutions. All of the solutions are represented by the equation

1

π

−π

−1

4π x

y = sin 3x

»

Try Exercise 66, page 279

EXAMPLE 5

Solve sin2 2x

»

Solve a Trigonometric Equation

兹3 兹3 sin 2x sin 2x 苷 0, where 0 x 360 . 2 2

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

Factor the left side of the equation by grouping, and then set each factor equal to zero.

The solutions are the x-coordinates of the x-intercepts of

sin2 2x

冉

兹3 兹3 sin 2x sin 2x 苷0 2 2

sin 2x sin 2x

冊冉冉

冊冊

兹3 兹3 sin 2x 苷0 2 2

共sin 2x 1兲 sin 2x sin 2x 1 苷 0 sin 2x 苷 1

or

兹3 苷0 2

sin 2x

兹3 苷0 2

sin 2x 苷

兹3 2

y 苷 sin2 2x

兹3 sin 2x 2

sin 2x

兹3 2

on the interval 关0, 2兲. See the following figure.

3.6 The equation sin 2x 苷 1 implies that 2x 苷 270 360 k, k an integer. Thus x 苷 135 180 k. The solutions of this equation with 兹3 0 x 360 are 135° and 315°. Similarly, the equation sin 2x 苷 2 implies 2x 苷 60 360 k x 苷 30 180 k

or

273

Trigonometric Equations y

( π3 , 0)

1

( 4π3 , 0)

( 3π4 , 0)

( 7π4 , 0) π

2x 苷 120 360 k x 苷 60 180 k

The solutions with 0 x 360 are 30°, 60°, 210°, and 240°. Combining the solutions from each equation, we have 30°, 60°, 135°, 210°, 240°, and 315° as our solutions.

−1

( π6 , 0)

2π

x

( 7π6 , 0)

»

Try Exercise 84, page 279

In Example 6, algebraic methods do not provide the solutions, so we rely on a graph.

EXAMPLE 6

»

Approximate Solutions Graphically

Use a graphing utility to approximate the solutions of x 3 cos x 苷 0. 3.5

Solution The solutions are the x-intercepts of y 苷 x 3 cos x. See Figure 3.26. A closeup view of the graph of y 苷 x 3 cos x shows that, to the nearest thousandth, the solutions are

(2.938, 0) (2.663, 0) − 1.5

(−1.170, 0) − 0.5

y 苷 x 3 cos x

x1 苷 1.170,

3.5

x2 苷 2.663,

and x3 苷 2.938

»

Try Exercise 86, page 279

Figure 3.26

EXAMPLE 7

»

Solve a Projectile Application

A projectile is fired at an angle of inclination from the horizon with an initial velocity v0. Its range d (neglecting air resistance) is given by d苷

共v 0 兲2 sin u cos u 16

Path of projectile

where v0 is measured in feet per second and d is measured in feet. See Figure 3.27. θ d

x

Figure 3.27 Continued 䉴

274

Chapter 3

Trigonometric Identities and Equations a.

If v0 苷 325 feet per second, find the angles (in degrees) for which the projectile will hit a target 2295 feet downrange.

b.

What is the maximum horizontal range for a projectile that has an initial velocity of 474 feet per second?

c.

Determine the angle of inclination that produces the maximum range.

Solution a.

We need to solve 2295 苷

3252 sin cos 16

(1)

for , where 0 90 . The following solutions were obtained by using a graphing 3252 utility to graph d 苷 2295 and d 苷 sin cos . See Figure 3.28. Thus 16 there are two angles for which the projectile will hit the target. To the nearest thousandth of a degree, they are Method 1

苷 22.025

and

苷 67.975

It should be noted that the graph in Figure 3.28 is not a graph of the path of the projectile. It is a graph of the distance d as a function of the angle . 4000

d=

3252 sin θ cos θ 16

Method 2 To solve algebraically, we proceed as follows. Multiply each side of Equation (1) by 16 and divide by 3252 to produce

d, in feet

d = 2295

(22.025, 2295)

sin cos 苷

(67.975, 2295) 0

100 0

共16兲共2295兲 3252

The identity 2 sin cos 苷 sin 2 gives us sin cos 苷

θ , in degrees

Figure 3.28

Hence

sin 2 . 2

sin 2 共16兲共2295兲苷 2 3252 共16兲共2295兲 sin 2 苷 2 ⬇ 0.69529 3252

There are two angles in the interval 关0 , 180 兴 whose sines are 0.69529. One is sin1 0.69529, and the other is the reference angle for sin1 0.69529. Therefore, 2u ⬇ sin1 0.69529 1 u ⬇ sin1 0.69529 2 u ⬇ 22.025

or

2u ⬇ 180 sin1 0.69529 1 u ⬇ 共180 sin1 0.69529兲 2 u ⬇ 67.975

These are the same angles that we obtained using Method 1.

3.6 8000

Use a graphing utility to find that the graph of d 苷

b.

(45, 7021.125)

275

Trigonometric Equations

4742 sin cos has a 16

d, in feet

maximum value of d 苷 7021.125 feet. See Figure 3.29. In b., the maximum value is attained for u 苷 45 .

c. 0

100 0

θ , in degrees

d苷

»

Try Exercise 92, page 279

4742 sin cos 16

Figure 3.29

Model Sinusoidal Data Data that can be closely modeled by a function of the form y 苷 a sin 共bx c兲 d are called sinusoidal data. Many graphing utilities are designed to perform a sine regression to find the sine function that provides the best least-squares fit for sinusoidal data. For instance, the TI-83/TI-83 Plus/TI-84 Plus uses the command SinReg and the TI-86 uses SinR to perform a sine regression. The process generally works best for those sets of data for which we have a good estimate of the period. In Example 8 we use a sine regression to model the percent of illumination of the moon as seen from Earth. See Figure 3.30.

Figure 3.30

EXAMPLE 8

»

Use a Sine Regression to Model Data

Table 3.3 on the following page shows the percent of the moon illuminated, at midnight Central Standard Time, for selected days of January 2009. Find the regression function that models the percent of the moon illuminated as a function of the day number, with January 1, 2009 represented by x 苷 1. Use the function to estimate the percent of the moon illuminated at midnight Central Standard Time on January 21, 2009. Note: The illumination cycle of the moon has a period of about 29.53 days.

Continued 䉴

276

Chapter 3

Trigonometric Identities and Equations Table 3.3 Midnight of: Date in 2009 (CST)

Day Number

% of Moon Illuminated

3

37

Jan. 3 Jan. 7

7

79

Jan. 11

11

100

Jan. 15

15

78

Jan. 19

19

39

Jan. 23

23

8

Jan. 27

27

1

Jan. 31

31

22

The percent of the moon illuminated is nearly the same regardless of one’s position on Earth. Source: The Astronomical Applications Department of the U.S. Naval Observatory.

TO REVIEW

Solution 1.

How to Construct a Scatter Plot See section 1.7.

Construct a scatter plot of the data. Enter the data from Table 3.3 into your graphing utility. See Figure 3.31. The sinusoidal nature of the scatter plot in Figure 3.32 indicates that the data can be effectively modeled by a sine function. 100

L1 3 7 11 15 19 23 27 L1(1) = 3

L2 37 79 100 78 39 8 1

L3 1 ––––––

0

40 0

Figure 3.31

Integrating Technology A sine regression requires at least four data points. The iterations number 16 in Figure 3.33 is the maximum number of times the SinReg command will iterate to find a regression equation. Any integer from 1 to 16 can be used; however, the integer 16 generally produces more accurate results than smaller integers.

2.

Figure 3.32

Find the regression equation. On a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, the input shown in Figure 3.33 produces the results in Figure 3.34. Regression equation is stored in Y1

SinReg 16, L1, L2, 29.53, Y1 Iterations x-values in L1

Period y-values in L2

Figure 3.33

SinReg y=a*sin(bx+c)+d a=49.59853514 b=.2129362398 c=-.7940588366 d=46.72225222

Figure 3.34

The regression equation is y ⬇ 49.60 sin共0.2129x 0.7941兲 46.72.

3.6

Trigonometric Equations

277

Examine the fit. The SinReg command does not yield a correlation coefficient. However, a graph of the regression equation and the scatter plot of the data shows that the regression equation provides a good model. See Figure 3.35.

3.

100

40

0 0

Figure 3.35

According to the following result, the percent of the moon illuminated at midnight (CST) on January 21, 2009 共x 苷 21兲 will be about 21%. y ⬇ 49.60 sin共0.2129共21兲 0.7941兲 46.72 ⬇ 21

»

Try Exercise 94, page 280

Topics for Discussion 1. A student finds that x 苷 0 is a solution of sin x 苷 x. Because the function y 苷 sin x has a period of 2, the student reasons that 2, 4, 6, . . . are also solutions. Explain why the student is not correct.

冉冊

2. How many solutions does the equation 2 sin x 0 x 2? Explain.

3. How many solutions does the equation sin 0x

2

苷 5 have on the interval

1 苷 0 have on the interval x

? Explain. 2

1 has solutions of x 苷 2 6 5 1 and x 苷 . How would you write the solutions of sin x 苷 if the real 6 2 number x were not restricted to the interval 关0, 2兲?

4. On the interval 0 x 2, the equation sin x 苷

278

Chapter 3

Trigonometric Identities and Equations

Exercise Set 3.6 In Exercises 1 to 22, solve each equation for solutions in the interval 0 x 2 .

11. sec x 兹2 苷 0

12. 2 sin x 苷兹3

13. tan x 兹3 苷 0

14. cos x 1 苷 0

34.

1 1 1 2 cos x 苷 cos x 5 2 3 2

35. 3 tan2 x 2 tan x 苷 0

36. 4 cot 2 x 3 cot x 苷 0

37. 3 cos x sec x 苷 0

38. 5 sin x csc x 苷 0

15. 2 sin x cos x 苷兹2 cos x

39. tan2 x 苷 3 sec2 x 2

16. 2 sin x cos x 苷兹3 sin x

40. csc2 x 1 苷 3 cot 2 x 2

17. sin2 x 1 苷 0

18. cos2 x 1 苷 0

41. 2 sin2 x 苷 1 cos x

19. 4 sin x cos x 2 兹3 sin x 2 兹2 cos x 兹6 苷 0

42. cos2 x 4 苷 2 sin x 3

10. sec2 x 兹3 sec x 兹2 sec x 兹6 苷 0

43. 3 cos2 x 5 cos x 2 苷 0

11. csc x 兹2 苷 0

12. 3 cot x 兹3 苷 0

44. 2 sin2 x 5 sin x 3 苷 0

13. 2 sin2 x 1 苷 3 sin x

14. 2 cos2 x 1 苷 3 cos x

45. 2 tan2 x tan x 10 苷 0

15. 4 cos2 x 3 苷 0

16. 2 sin2 x 1 苷 0

46. 2 cot 2 x 7 cot x 3 苷 0

17. 2 sin3 x 苷 sin x

18. 4 cos3 x 苷 3 cos x

47. 3 sin x cos x cos x 苷 0

19. 4 sin2 x 2 兹3 sin x 兹3 苷 2 sin x 20. tan2 x tan x 兹3 苷兹3 tan x 21. sin4 x 苷 sin2 x

22. cos4 x 苷 cos2 x

48. tan x sin x sin x 苷 0 49. 2 sin x cos x sin x 2 cos x 1 苷 0 50. 6 cos x sin x 3 cos x 4 sin x 2 苷 0 51. 2 sin x cos x 苷 1

In Exercises 23 to 60, solve each equation, where 0° x 360°. Round approximate solutions to the nearest tenth of a degree.

23. cos x 0.75 苷 0

24. sin x 0.432 苷 0

25. 3 sin x 5 苷 0

26. 4 cos x 1 苷 0

27. 3 sec x 8 苷 0

28. 4 csc x 9 苷 0

29. cos x 3 苷 0

30. sin x 4 苷 0

31. 3 5 sin x 苷 4 sin x 1 32. 4 cos x 5 苷 cos x 3 33.

1 2 3 3 sin x 苷 sin x 2 3 4 5

52. sin x 2 cos x 苷 1 53. 2 sin x 3 cos x 苷 1 54. 兹3 sin x cos x 苷 1 55. 3 sin2 x sin x 1 苷 0 56. 2 cos2 x 5 cos x 5 苷 0 57. 2 cos x 1 3 sec x 苷 0 58. 3 sin x 5 csc x 苷 0 59. cos2 x 3 sin x 2 sin2 x 苷 0 60. sin2 x 苷 2 cos x 3 cos2 x

3.6 In Exercises 61 to 70, find the exact solutions, in radians, of each trigonometric equation.

2 兹3 苷0 3

61. tan 2x 1 苷 0

62. sec 3x

63. sin 5x 苷 1

64. cos 4x 苷

65. sin 2x sin x 苷 0

冉

p 67. sin 2x 6 69. sin2

冊

1 苷 2

x cos x 苷 1 2

66. cos 2x 苷

冉

兹2 2 兹3 2

p 68. cos 2x 4 70. cos2

冊

苷

兹2 2

71. cos 2x 苷 1 3 sin x

72. cos 2x 苷 2 cos x 1

73. sin 4x sin 2x 苷 0

74. sin 4x cos 2x 苷 0

x 苷 sin x 2

76. tan

88. cos x 苷

x 苷 1 cos x 2

89.

Use a graphing utility to solve cos x 苷 x 3 x by graphing each side and finding the x value of each point of intersection. Round to the nearest hundredth.

90.

Approximate the largest value of k for which the equation sin x cos x 苷 k has a solution.

PROJECTILES Exercises 91 and 92 make use of the following. A projectile is fired at an angle of inclination from the horizon with an initial velocity v0. Its range d (neglecting air resistance) is given by d

79. sin x cos 2x cos x sin 2x 苷

兹3 2

(v0)216 sin cos

where v0 is measured in feet per second and d is measured in feet.

91.

If v0 苷 288 feet per second, use a graphing utility to find the angles u (to the nearest hundredth of a degree) for which the projectile will hit a target 1295 feet downrange.

92.

Find the maximum horizontal range, to the nearest tenth of a foot, for a projectile that has an initial velocity of 375 feet per second. What value of u produces this maximum horizontal range?

93.

SUNRISE TIME The table below shows the sunrise time for Atlanta, Georgia, for selected days

77. sin 2x cos x cos 2x sin x 苷 0 78. cos 2x cos x sin 2x sin x 苷 0

in 2008. Date

Day Number

Sunrise Time

80. cos 2x cos x sin 2x sin x 苷 1

Jan. 1

1

7:42

81. sin 3x sin x 苷 0

Feb. 1

32

7:35

Mar. 1

61

7:06

April 1

92

6:25

May 1

122

5:48

June 1

153

5:28

July 1

183

5:31

Aug. 1

214

5:50

Sept. 1

245

6:12

Oct. 1

275

6:32

Nov. 1

306

6:57

Dec. 1

336

7:25

82. cos 3x cos x 苷 0

83. 2 sin x cos x 2 sin x cos x 1 苷 0 84. 2 sin x cos x 2 兹2 sin x 兹3 cos x 兹6 苷 0 In Exercises 85 to 88, use a graphing utility to solve the equation. State each approximate solution accurate to the nearest ten-thousandth.

85. cos x 苷 x, where 0 x 2p 86. 2 sin x 苷 x, where 0 x 2p 87. sin 2x 苷

1 , where 4 x 4 x

279

1 , where 0 x 5 x

x cos x 苷 1 2

In Exercises 71 to 84, solve each equation for exact solutions in the interval 0 x 2.

75. tan

Trigonometric Equations

Source: The U.S. Naval Observatory. Note: The times do not reflect daylight savings time.

280

Chapter 3

Trigonometric Identities and Equations

a. Find the sine regression function that models the sunrise time, in hours, as a function of the day number. Let x 苷 1 represent January 1, 2008. Assume that the sunrise times have a period of 365.25 days. b. Use the regression function to estimate the sunrise time (to the nearest minute) for March 11, 2008 共x 苷 71兲. SUNSET TIME The table below shows the sunset time for Omaha, Nebraska, for selected days

94. in 2009.

Date

Day Number

Sunset Time

Jan. 1

1

17:06

Feb. 1

32

17:41

Mar. 1

60

18:15

April 1

91

18:49

May 1

121

19:22

June 1

152

19:51

July 1

182

20:01

Aug. 1

213

19:40

Sept. 1

244

18:56

Oct. 1

274

18:05

Nov. 1

305

17:19

Dec. 1

335

16:56

95.

PERCENT OF THE MOON ILLUMINATED The table below shows the percent of the moon illuminated at midnight, Central Standard Time, for selected days in October and November of 2008.

Midnight of: Date in 2008

Day Number

% of Moon Illuminated

Oct. 1

1

4

Oct. 5

5

31

Oct. 9

9

68

Oct. 13

13

97

Oct. 17

17

92

Oct. 21

21

53

Oct. 25

25

14

Oct. 29

29

0

Nov. 2

33

16

Nov. 6

37

51

Nov. 10

41

88

Nov. 14

45

98

Nov. 18

49

68

Nov. 22

53

26

Nov. 26

57

2

Nov. 30

61

6

Source: The U.S. Naval Observatory. Source: The U.S. Naval Observatory. Note: The times do not reflect daylight savings time.

a. Find the sine regression function that models the sunset time, in hours, as a function of the day number. Let x 苷 1 represent January 1, 2009. Assume that the sunset times have a period of 365.25 days. b. Use the regression function to estimate the sunset time (to the nearest minute) for March 14, 2009 共x 苷 73兲.

a. Find the sine regression function that models the percent of the moon illuminated as a function of the day number. Let x 苷 1 represent October 1, 2008. Use 29.53 days for the period of the data. b. Use the regression function to estimate the percent of the moon illuminated (to the nearest 1 percent) at midnight Central Standard Time on October 31, 2008.

3.6 96.

HOURS OF DAYLIGHT The table below shows the hours of daylight for Dallas, Texas, for selected days in 2009.

Date

Day Number

Time of day

Altitude (degrees)

6:00

9.9

7:00

1.4

Hours of Daylight (hours:minutes)

8:00

11.5

9:00

21.0

Jan. 1

1

10:03

10:00

29.0

Feb. 1

32

10:38

11:00

34.7

Mar. 1

60

11:30

12:00

37.5

April 1

91

12:33

13:00

36.7

May 1

121

13:29

14:00

32.6

June 1

152

14:11

15:00

25.7

July 1

182

14:16

16:00

17.0

Aug. 1

213

13:44

17:00

7.2

Sept. 1

244

12:50

18:00

3.6

Oct. 1

274

11:50

Nov. 1

305

10:51

Dec. 1

335

10:09

Source: Data extracted from sunrise-sunset times given on the World Wide Web by the U.S. Naval Observatory.

Source: The U.S. Naval Observatory.

98.

MODEL THE DAYLIGHT HOURS For a particular day of the year t, the number of daylight hours in New Orleans can be approximated by

冉

d共t兲苷 1.792 sin a. Find the sine regression function that models the hours of daylight as a function of the day number. Let x 苷 1 represent January 1, 2009. Use 365.25 for the period of the data. b. Use the regression function to estimate the hours of daylight (stated in hours and minutes, with the minutes rounded to the nearest minute) for Dallas on May 12, 2009. 97.

ALTITUDE OF THE SUN The table at the top of the next column shows the altitude of the sun for Detroit, Michigan, at selected times during October 19, 2007. a. Find the sine regression function that models the altitude, in degrees, of the sun as a function of the time of day. Use 24.03 hours (the time from sunrise October 19 to sunrise October 20) for the period. b. Use the regression function to estimate the altitude of the sun (to the nearest 0.1 degree) on October 19, 2007, at 9:25.

281

Trigonometric Equations

冊

2p共t 80兲 12.145 365

where t is an integer and t 苷 1 corresponds to January 1. According to d, how many days per year will New Orleans have at least 10.75 hours of daylight? 99. CROSS-SECTIONAL AREA A rain gutter is constructed from a long sheet of aluminum that measures 9 inches in width. The aluminum is to be folded as shown by the cross section in the following diagram. a. Verify that the area of the cross section is given by A 苷 9 sin u共cos u 1兲, where 0 u 90 . b.

What values of u, to the nearest degree, produce a cross-sectional area of 10.5 square inches?

c.

Determine the value of u that produces the cross section with the maximum area. x

θ

θ y

3 in.

3 in.

θ

θ 3 in.

282

Chapter 3

Trigonometric Identities and Equations

100. OBSERVATION ANGLE A person with an eye level of 5 feet 6 inches is standing in front of a painting, as shown in the following diagram. The bottom of the painting is 3 feet above floor level, and the painting is 6 feet in height. The angle u shown in the figure is called the observation angle for the painting. The person is d feet from the painting.

冉

a. Verify that u 苷 tan1 b.

where u is the angle the bus driver has turned the front of the bus. Find the value of x for u 苷 20 and u 苷 30 . Round to the nearest hundredth of a foot.

冊

θ

6d . d 8.75 2

Find the distance d, to the nearest tenth of a foot, p for which u 苷 . 6

B x

B

A1

A2

6 ft

θ

3 ft d

102. OPTIMAL BRANCHING OF BLOOD VESSELS It is hypothesized that the system of blood vessels in primates has evolved so that it has an optimal structure. In the case of a blood vessel splitting into two vessels, as shown in the accompanying figure, we assume that both new branches carry equal amounts of blood. A model of the angle u is given by the equation cos u 苷 2共x4兲/共x4兲. The value of x is such that 1 x 2 and depends on assumptions about the thickness of the blood vessels. Assuming this is an accurate model, find the range of values of the angle u.

101. MODEL THE MOVEMENT OF A BUS As bus A1 makes a left turn, the back B of the bus moves to the right. If bus A2 were waiting at a stoplight while A1 turned left, as shown in the following figure, there is a chance the two buses would scrape against one another. For a bus 28 feet long and 8 feet wide, the movement of the back of the bus to the right can be approximated by

θ θ

x 苷兹共4 18 cot u兲2 100 共4 18 cot u兲

Connecting Concepts In Exercises 103 to 108, solve each equation for exact solutions in the interval 0 x 2.

106. 兹3 sin x cos x 苷 1

103. 兹3 sin x cos x 苷兹3

107. cos 5x cos 3x 苷 0

104. sin x cos x 苷 1

108. cos 5x cos x sin 3x 苷 0

105. sin x 兹3 cos x 苷兹3

Exploring Concepts with Technology

283

Projects

May 1

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

Tethys

Enceladus

Titan 9

8

7

10 Rhea

Dione 6

5

4

EAST

3

2

Apr. 1

WEST

1. THE MOONS OF SATURN The accompanying figure shows the east-west displacement of five moons of Saturn. The figure shows that the period of the sine curve that models the displacement of the moon Titan is about 15.95 Earth days. The period of the sine curve that models the displacement of the moon Rhea is about 4.52 Earth days.

Source: “Saturn’s Satellites” from Sky & Telescope, April 2006, p. 56. Copyright 2006 by Sky Publishing Company. Reprinted with permission.

a. Find a sine regression function that models the displacement of the moon Titan. Use x 苷 0 to represent the beginning of April 1, 2006, and think of “east” as a negative displacement and “west” as a positive displacement. Use 1 for the amplitude. b. Work a. for the moon Rhea. Use 0.4 for the amplitude. c. An astronomer viewed the moons Titan and Rhea at 11:59 P.M. on May 10, 2006 (x 苷 40). Were these moons on opposite sides of Saturn or on the same (east/west) side at that time?

Exploring Concepts with Technology Approximate an Inverse Trigonometric Function with Polynomials The function y 苷 sin1 x can be approximated by polynomials. For example, consider the following: f1共x兲苷 x

x3 23

f2共x兲苷 x

x3 1 3x 5 23 245

f3共x兲苷 x

x3 1 3x 5 1 3 5x 7 23 245 2467

where 1 x 1 where 1 x 1 where 1 x 1

284

Chapter 3

Trigonometric Identities and Equations

f4共x兲苷 x

1 3x 5 1 3 5x 7 1 3 5 7x 9 x3 23 245 2467 24689

where 1 x 1

fn共x兲苷 x

x3 1 3x5 1 3 5x7 共2n兲! x2n1 n 2 23 245 2467 共2 n!兲共2n 1兲

where 1 x 1, n! 苷 1 2 3 共n 1兲n and

共2n兲! 苷 1 2 3 共2n 1兲共2n兲

Use a graphing utility for the following exercises. 1. Graph y 苷 f1共x兲, y 苷 f2共x兲, y 苷 f3共x兲, and y 苷 f4共x兲 in the viewing window Xmin 1, Xmax 1, Ymin 1.5708, Ymax 1.5708. 2. Determine the values of x for which f3共x兲 and sin1 x differ by less than 0.001. That is, determine the values of x for which 兩 f3共x兲 sin1 x兩 0.001. 3. Determine the values of x for which 兩 f4共x兲 sin1 x兩 0.001. 4. Write all seven terms of f6共x兲. Graph y 苷 f6共x兲 and y 苷 sin1 x on the viewing

window Xmin 1, Xmax 1, Ymin , Ymax . 2 2 5. Write all seven terms of f6共1兲. What do you notice about the size of a term compared to that of the preceding term? 6. What is the largest-degree term in f10共x兲?

Chapter 3 Summary 3.1 Verification of Trigonometric Identities • Trigonometric identities are verified by using algebraic methods and previously proved identities. • The Fundamental Trigonometric Identities are given in Table 3.1, page 217. • Guidelines for verifying trigonometric identities are given on pages 217 and 218.

• Sum and difference identities for the sine function sin共a b兲苷 sin a cos b cos a sin b sin共a b兲苷 sin a cos b cos a sin b • Sum and difference identities for the tangent function tan共a b兲苷

tan a tan b 1 tan a tan b

tan共a b兲苷

tan a tan b 1 tan a tan b

3.2 Sum, Difference, and Cofunction Identities • Sum and difference identities for the cosine function cos共a b兲苷 cos a cos b sin a sin b cos共a b兲苷 cos a cos b sin a sin b

• Cofunction identities with u in degrees sin共90 u兲苷 cos u

cos共90 u兲苷 sin u

tan共90 u兲苷 cot u

cot共90 u兲苷 tan u

sec共90 u兲苷 csc u

csc共90 u兲苷 sec u

If u is in radian measure, replace 90° with

p . 2

Summary 3.3 Double- and Half-Angle Identities

• Sum-to-product identities sin x sin y 苷 2 sin

xy xy cos 2 2

cos x cos y 苷 2 cos

xy xy cos 2 2

苷 2 cos2 a 1

sin x sin y 苷 2 cos

xy xy sin 2 2

2 tan a 1 tan2 a

cos x cos y 苷 2 sin

• Double-angle identities sin 2a 苷 2 sin a cos a cos 2a 苷 cos2 a sin2 a 苷 1 2 sin2 a

tan 2a 苷

• Power-reducing identities

a sin x b cos x 苷 k sin共x 兲

cos2 a 苷

1 cos 2a 2

where k 苷兹a 2 b 2, sin 苷

tan2 a 苷

1 cos 2a 1 cos 2a

cos 苷

冑冑冑

sin

a 苷

2

1 cos a 2

cos

a 苷

2

1 cos a 2

tan

a 苷

2

1 cos a sin a 1 cos a 苷苷 1 cos a 1 cos a sin a

The choice of the plus or minus sign depends on the a quadrant in which lies. 2

3.4 Identities Involving the Sum of Trigonometric Functions • Product-to-sum identities sin a cos b 苷

1 关sin共a b兲 sin共a b兲兴 2

cos a sin b 苷

1 关sin共a b兲 sin共a b兲兴 2

cos a cos b 苷

1 关cos共a b兲 cos共a b兲兴 2

1 sin a sin b 苷关cos共a b兲 cos共a b兲兴 2

xy xy sin 2 2

• For sums of the form a sin x b cos x,

1 cos 2 a sin2 a 苷 2

• Half-angle identities

285

a 兹a2 b 2

b 兹a 2 b 2

, and

.

3.5 Inverse Trigonometric Functions • The inverse of y 苷 sin x is y 苷 sin1 x, with 1 x 1 and y . 2 2 • The inverse of y 苷 cos x is y 苷 cos1 x, with 1 x 1 and 0 y . • The inverse of y 苷 tan x is y 苷 tan1 x, with x p p . and y 2 2 • The inverse of y 苷 cot x is y 苷 cot1 x, with x and 0 y . • The inverse of y 苷 csc x is y 苷 csc1 x, with x 1 or x 1 p p , y 苷 0. and y 2 2 • The inverse of y 苷 sec x is y 苷 sec1 x, with x 1 or x 1 p . and 0 y p, y 苷 2

3.6 Trigonometric Equations • Algebraic methods and identities are used to solve trigonometric equations. Because the trigonometric functions are periodic, there may be an infinite number of solutions. If solutions cannot be found by algebraic methods, then we often use a graphing utility to find approximate solutions.

286

Chapter 3

Trigonometric Identities and Equations

Chapter 3 Assessing Concepts 1. True or False: cos2 a 苷

1 cos 2 a is an identity. 2

2. True or False: For all real numbers x, cos1(cos x) 苷 x. 3. True or False: For all real numbers x, cos (cos1 x) 苷 x. 4. True or False: The graph of y 苷 sin1 x is symmetric with respect to the origin. 5. How many solutions does the equation sin 2x 苷 0.3 have on the interval 0 x 2p?

6. What is the domain of f (x) 苷 cos1 2x? 7. What is the range of y 苷 cos1 x?

冉冊

8. Determine the exact value of sin1 sin

7p . 3

9. What is the amplitude of f共x兲苷 sin x cos x? 10. Determine the exact value of tan 75 .

Chapter 3 Review Exercises In Exercises 1 to 10, find the exact value.

11. cos 共45 30 兲

冉

13. sin

2 3 4

12. tan 共210 45 兲

冊

冉冉

14. sec

15. sin共60 135 兲

4 3 4

17. sin 22

1 2

18. cos 105

19. tan 67

1 2

10. sin 112.5

a. sin共兲

b. tan 2

冉冊

c. cos

2

兹2 兹3 with 0° a 90°, and cos b 苷 2 2 with 270° b 360°, find

14. Given sin a 苷

b. tan 2

c. sin 2

In Exercises 15 to 20, write the given expression as a single trigonometric function.

1 1 with 0° a 90°, and cos b 苷 with 2 2 270° b 360°, find

11. Given sin a 苷

b. tan 2

冉冊

c. sin

2

1 兹3 with 90° a 180°, and cos b 苷 2 2 with 180° b 270°, find

12. Given sin a 苷

a. sin共兲

1 with 270° a 360°, and 2

兹3 with 180° b 270°, find 2

a. cos共兲

In Exercises 11 to 14, find the exact values of the given functions.

a. cos共兲

cos b 苷

5 7 3 4

16. cos

冉冊冉冊

冊冊

13. Given sin a 苷

b. sec 2

冉冊

c. cos

2

15. 2 sin 3x cos 3x

16.

tan 2x tan x 1 tan 2x tan x

20.

1 cos 2 sin 2

17. sin 4x cos x cos 4x sin x 18. cos2 2 sin2 2 19.

sin 2 cos 2

In Exercises 21 to 24, write each expression as the product of two trigonometric functions.

21. cos 2u cos 4u

22. sin 3u sin 5u

23. sin 6 sin 2

24. sin 5 sin

Review Exercises

冉

In Exercises 25 to 42, verify the identity.

25.

1 1 苷 2 tan x sec x sin x 1 sin x 1

26.

sin x 苷 csc x cot x, 1 cos x

27.

1 sin x 苷 tan2 x 1 tan x sec x cos2 x

28. cos2 x sin2 x sin 2x 苷

29.

0x

46. cos 2 sin1

47. 2 sin1共x 1兲苷

cos2 2x sin2 2x cos 2x sin 2x

In Exercises 49 and 50, find all solutions of each equation with 0° x 360°.

49. 4 sin2 x 2 兹3 sin x 2 sin x 兹3 苷 0

冊

兹2 苷共cos sin 兲 4 2

32. sin共180 兲苷 sin cos cos sin 33.

3

4 苷 5 2

48. sin1 x cos1

30. sin共270 兲 cos共270 兲苷 sin cos

冉

冊

In Exercises 47 and 48, solve each equation.

2

1 cos x 苷 tan x sin x cos x

31. sin

3 5

287

sin 4x sin 2x 苷 cot 3x cos 4x cos 2x

50. 2 sin x cos x 兹2 cos x 2 sin x 兹2 苷 0 In Exercises 51 and 52, solve the trigonometric equation where x is in radians. Round approximate solutions to four decimal places.

51. 3 cos2 x sin x 苷 1 52. tan2 x 2 tan x 3 苷 0

34. 2 sin x sin 3x 苷共1 cos 2x兲共1 2 cos 2x兲

In Exercises 53 and 54, solve each equation on 0 x 2.

35. sin x cos 2x 苷共2 sin x 1兲共sin x 1兲

53. sin 3x cos x cos 3x sin x 苷

冉

36. cos 4x 苷 1 8 sin2 x 8 sin4 x

54. cos 2x

4 tan x 4 tan3 x 37. tan 4x 苷 1 6 tan2 x tan4 x 38.

3

冊

苷

兹3 2

In Exercises 55 to 58, write the equation in the form y k sin(x ), where the measure of is in radians. Graph one period of each function.

1 cos x sin 2x sin x 苷 cos 2x cos x sin x

39. 2 cos 4x sin 2x 苷 2 sin 3x cos 3x 2 sin x cos x

55. f 共x兲苷兹3 sin x cos x

40. 2 sin x sin 2x 苷 4 cos x sin2 x

56. f 共x兲苷 2 sin x 2 cos x

41. cos共x y兲 cos共x y兲苷 cos2 x cos2 y 1

57. f 共x兲苷 sin x 兹3 cos x

42. cos共x y兲 sin共x y兲苷 sin x cos x sin y cos y In Exercises 43 to 46, evaluate each expression.

冉冊冋冉冊

43. sec sin1

12 13

45. cos sin1

3 5

冉

44. cos sin1

cos1

1 2

册

5 13

3 5

冊

58. f 共x兲苷

1 兹3 sin x cos x 2 2

In Exercises 59 to 62, graph each function.

59. f 共x兲苷 2 cos1 x 61. f 共x兲苷 sin1

x 2

60. f 共x兲苷 sin1共x 1兲 62. f共x兲苷 sec1 2x

288 63.

Chapter 3

Trigonometric Identities and Equations a. Find the sine regression function that models the sunrise time, in hours, as a function of the day number. Let x 苷 1 represent January 1, 2009. Assume that the sunrise times have a period of 365.25 days.

SUNRISE TIME The table below shows the sunrise time for Flagstaff, Arizona, for selected days in 2009. Day Number

Date

Sunrise Time (hours:minutes)

Jan. 1

1

7:35

Feb. 1

32

7:25

Mar. 1

60

6:56

April 1

91

6:13

May 1

121

5:35

June 1

152

5:13

July 1

182

5:16

Aug. 1

213

5:36

Sept. 1

244

5:59

Oct. 1

274

6:22

Nov. 1

305

6:48

Dec. 1

335

7:17

b. Use the regression function to estimate the sunrise time (to the nearest minute) for April 14, 2009 共x 苷 104兲.

Source: The U.S. Naval Observatory. Note: The times are Mountain Standard Times.

» » » Quantitative Reasoning: Basketball and Trigonometric Equations »»» QR1. When a basketball player shoots at the basket, there are several factors that determine the horizontal distance the basketball travels. One equation for the distance s, in feet, is given by s苷 α

β s

v 2 cos a sec b sin共a b兲 16

(1)

where the angles a and b are as shown in the diagram at the left and v is the velocity in feet per second at which the ball leaves the player’s hand. p , v 苷 28 feet per second, and b a 1.35. 18 Use the graph to find the value of a that maximizes s for the given values.

a. Graph Equation (1) for b 苷

b. Show that the value of a in a. is approximately

p b . 4 2

Test

289

QR2. Verify the identity cos a sec b sin共a b兲苷 cos2 a共tan a tan b兲 and then use the identity to show that Equation (1) can be written as v 苷 4 sec a

冑

s tan a tan b

(2)

Note: Equation (2) gives the velocity at which a ball must be launched for various launch angles to reach a basket s feet away. p , s 苷 15 feet (the distance of a free throw), 18 and b a 1.35. Use the graph to find the value of a that maximizes v for the given values.

a. Graph Equation (2) for b 苷

p b . Note: The 4 2 significance of this result is that the ball must be launched at an angle of p b a苷 to travel the required distance with the least effort. 4 2

b. Show that the value of a in part a. is approximately

Chapter 3 Test 11. Verify the identity 1 sin2 x sec2 x 苷 sec2 x. 12. Verify the identity

19. Find the exact value of cos 2u given that sin 苷

4 and u 5

is in Quadrant II.

1 1 苷 2 tan x sec x tan x sec x tan x 13. Verify the identity cos3 x cos x sin2 x 苷 cos x. 14. Verify the identity csc x cot x 苷

1 cos x . sin x

15. Find the exact value of sin 195 . 3 兹2 , in Quadrant III, and cos 苷 , 5 2 in Quadrant II, find sin 共兲.

16. Given sin 苷

冉冊

17. Verify the identity sin

3 苷 cos . 2

18. Write cos 6x sin 3x sin 6x cos 3x in terms of a single trigonometric function.

10. Verify the identity tan

cos 苷 csc . 2 sin

11. Verify the identity sin2 2x 4 cos4 x 苷 4 cos2 x. 12. Find the exact value of sin 15 cos 75 . 1 兹3 sin x cos x in the form 2 2 y 苷 k sin共x 兲, where is measured in radians.

13. Write y 苷

14. Use a calculator to approximate the radian measure of cos1 0.7644 to the nearest thousandth.

冉

15. Find the exact value of sin cos1 16. Graph y 苷 sin1共x 2兲.

冊

12 . 13

290

Chapter 3

Trigonometric Identities and Equations

17. Solve 3 sin x 2 苷 0, where 0 x 360 . (State solutions to the nearest tenth of a degree.) 18. Solve sin x cos x

兹3 sin x 苷 0, where 0 x 2. 2

19. Find the exact solutions of sin 2x sin x 2 cos x 1 苷 0, where 0 x 2.

a. Find the sine regression function that models the hours of daylight as a function of the day number. Let x 苷 1 represent January 1, 2009. Use 365.25 for the period of the data. b. Use the regression function to estimate the hours of daylight (stated in hours and minutes, with the minutes rounded to the nearest minute) for Tampa on March 16, 2009 (x 苷 75).

HOURS OF DAYLIGHT The table below shows the hours of daylight for Tampa, Florida, for selected days in 2009.

20.

Date

Day Number

Hours of Daylight

Jan. 1

1

10:24

Feb. 1

32

10:53

Mar. 1

60

11:37

April 1

91

12:28

May 1

121

13:15

June 1

152

13:48

July 1

182

13:53

Aug. 1

213

13:26

Sept. 1

244

12:41

Oct. 1

274

11:52

Nov. 1

305

11:04

Dec. 1

335

10:30

Source: The U.S. Naval Observatory

Cumulative Review Exercises 1. Solve: 2x 1 7

6. Convert 240° to radians.

2. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共x 1兲 2.

7. Convert

3. Explain how to use the graph of y 苷 f 共x兲 to produce the graph of y 苷 f 共x兲.

8. Evaluate: sin

4. Determine whether f 共x兲苷 x sin x is an even function or an odd function.

9. Evaluate: csc 60°

5. Find the inverse of f 共x兲苷

5x . x1

5 to degrees. 3

3

10. Find tan , given that is an acute angle and sin 苷

2 . 3

Cumulative Review Exercises

11. Determine the sign of cot given that

3 . 2

12. What is the measure of the reference angle for the angle 苷 310°? 13. What is the measure of the reference angle for the angle 5 苷 ? 3

冉冊

14. Find the x- and y-coordinates of the point defined by W , where W is the wrapping function. 3

冉

291

冊

15. Find the amplitude, the period, and the phase shift for the . graph of y 苷 0.43 sin 2x 6 1 16. Evaluate: sin1 2 17. Use a calculator to evaluate cos1共0.8兲. Round to the nearest thousandth. 18. Use interval notation to state the domain of f 共x兲苷 cos1 x. 19. Use interval notation to state the range of f 共x兲苷 tan1 x. 20. Find the exact solutions of 2 cos2 x 1 苷 sin x, where 0 x 2.

Applications of Trigonometry

4 Elizabeth City

Nags Head

North Carolina

10

5m

ile s

64.9°

New Bern 39.4° Morehead City

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

292

4.1

The Law of Sines

4.2

The Law of Cosines and Area

4.3

Vectors

Trigonometry and Indirect Measurement In Chapter 2 we used trigonometric functions to find the unknown length of a side of a given right triangle. In this chapter we develop theorems that can be used to find the length of a side or the measure of an angle of any triangle, even if it is not a right triangle. These theorems are often used in the areas of Hurricane navigation, surveying, and building design. Meterologists use these theorems to estimate the distance from an approaching hurricane to cities in the projected path of the hurricane. For instance, in the diagram on the left, the distance from the hurricane to Nags Head can be determined using the Law of Sines, a theorem presented in this chapter. See Exercises 30 and 31 on page 299 for additional applications that can be solved by using the Law of Sines.

4.1

The Law of Sines

Section 4.1 ■ ■

■

293

The Law of Sines

The Law of Sines The Ambiguous Case (SSA) Applications of the Law of Sines

The Law of Sines Solving a triangle involves finding the lengths of all sides and the measures of all angles in the triangle. In this section and the next we develop formulas for solving an oblique triangle, which is a triangle that does not contain a right angle. The Law of Sines can be used to solve oblique triangles in which either two angles and a side or two sides and an angle opposite one of the sides are known. In Figure 4.1, altitude CD is drawn from C. The length of the altitude is h. Triangles ACD and BCD are right triangles. Using the definition of the sine of an angle of a right triangle, we have from Figure 4.1

h a h 苷 a sin B

sin A 苷

sin B 苷 C

共1兲

h b

h 苷 b sin A

共2兲

Equating the values of h in Equations (1) and (2), we obtain b

a

h

A

D

c

a sin B 苷 b sin A Dividing each side of the equation by sin A sin B, we obtain B

a b 苷 sin A sin B

Figure 4.1

Similarly, when an altitude is drawn to a different side, the following formulas result: c b c a 苷 and 苷 sin C sin B sin C sin A

take note The Law of Sines may also be written as

The Law of Sines If A, B, and C are the measures of the angles of a triangle and a, b, and c are the lengths of the sides opposite those angles, then

sin A sin B sin C 苷苷 a b c

a b c 苷苷 sin A sin B sin C

EXAMPLE 1

»

Solve a Triangle Using the Law of Sines

Solve triangle ABC if A 苷 42 , B 苷 63 , and c 苷 18 centimeters. Continued

䉴

294

Chapter 4

Applications of Trigonometry Solution Find C by using the fact that the sum of the interior angles of a triangle is 180°. A B C 苷 180 42 63 C 苷 180 C 苷 75

take note

Use the Law of Sines to find a.

We have used the rounding con-

a c 苷 sin A sin C a 18 苷 • A 42°, c 18, C 75° sin 42 sin 75 18 sin 42 a苷 ⬇ 12 centimeters sin 75

ventions stated on page 140 to determine the number of significant digits to be used for a and b.

Use the Law of Sines again, this time to find b. b c 苷 sin B sin C b 18 苷 • B 63°, c 18, C 75° sin 63 sin 75 18 sin 63 b苷 ⬇ 17 centimeters sin 75

C 75°

A

63°

42° 18 cm

Figure 4.2

B

The solution is C 苷 75 , a ⬇ 12 centimeters, and b ⬇ 17 centimeters. A scale drawing can be used to see if these results are reasonable. See Figure 4.2.

»

Try Exercise 4, page 298

The Ambiguous Case (SSA) When you are given two sides of a triangle and an angle opposite one of them, you may find that the triangle is not unique. Some information may result in two triangles, and some may result in no triangle at all. It is because of this that the case of knowing two sides and an angle opposite one of them (SSA) is called the ambiguous case of the Law of Sines. Suppose that sides a and c and the nonincluded angle A of a triangle are known and we are asked to solve triangle ABC. The relationships among h, the height of the triangle, a (the side opposite ⬔A), and c determine whether there are no, one, or two triangles. Case 1 First consider the case in which ⬔A is an acute angle (see Figure 4.3). There are four possible situations.

1. a h; there is no possible triangle. 2. a 苷 h; there is one triangle, a right triangle. 3. h a c; there are two possible triangles. 4. a c; there is one triangle, which is not a right triangle.

4.1 B

B a

c

h

α

B

B c

a=h

a

h

295

The Law of Sines

c

a

a h

α

A

A

C

1. a h; no triangle

b

C

b

A

2. a 苷 h; one triangle

A

C′′

C′

3. h a c; two triangles

b

C

4. a c; one triangle

Figure 4.3 Case 1: A is an acute angle.

Case 2 Now consider the case in which ⬔A is an obtuse angle (see Figure 4.4). Here, there are two possible situations.

1. a c; there is no triangle. 2. a c; there is one triangle. B

B a h

h

c A

A

C

1. a c; no triangle

a

c

b

C

2. a c; one triangle

Figure 4.4 Case 2: A is an obtuse angle.

take note When you solve for an angle of a triangle by using the Law of Sines,

EXAMPLE 2

be aware that the value of the

»

Solve a Triangle Using the Law of Sines (SSA)

inverse sine function will give the

a.

Find A, given triangle ABC with B 苷 32 , a 苷 42, and b 苷 30.

measure of an acute angle. If the

b.

Find C, given triangle ABC with A 苷 57 , a 苷 15 feet, and c 苷 20 feet.

situation is the ambiguous case

Solution

(SSA), you must consider a second, obtuse angle by using the

a.

supplement of the angle. You can use a scale drawing to see whether your results are reasonable.

C

42 30

h

30

32° B

A

A

Figure 4.5

b a 苷 sin B sin A 30 42 苷 sin 32 sin A 42 sin 32 sin A 苷 ⬇ 0.7419 30 A ⬇ 48 or 132

• B 32°, a 42, b 30

• The two angles with measure between 0° and 180° that have a sine of 0.7419 are approximately 48° and 132°.

To check that A ⬇ 132 is a valid result, add 132° to the measure of the given angle B (32°). Because 132 32 180 , we know that A ⬇ 132 is a valid result. Thus angle A ⬇ 48 or A ⬇ 132 (⬔BAC in Figure 4.5). Continued

䉴

296

Chapter 4

Applications of Trigonometry

B

c = 20 ft

a c 苷 sin A sin C 15 20 苷 sin 57 sin C 20 sin 57 sin C 苷 ⬇ 1.1182 15

b.

a = 15 ft

h

57°

Because 1.1182 is not in the range of the sine function, there is no solution of the equation. Thus there is no triangle for these values of A, a, and c. See Figure 4.6.

C

A

• A 57°, a 15, c 20

Figure 4.6

»

Try Exercise 20, page 298

Applications of the Law of Sines EXAMPLE 3

»

Solve an Application Using the Law of Sines

A radio antenna 85 feet high is located on top of an office building. At a distance AD from the base of the building, the angle of elevation to the top of the antenna is 26°, and the angle of elevation to the bottom of the antenna is 16°. Find the height of the building.

Solution Sketch a diagram. See Figure 4.7. Find the measure of angle B and the measure of angle . B 苷 90 26 苷 64 苷 26 16 苷 10

B 85 ft

β

C h

16°

Because we know the length BC and the measure of , we can use triangle ABC and the Law of Sines to find length AC. BC AC 苷 sin sin B 85 AC 苷 sin 10 sin 64 85 sin 64 AC 苷 sin 10

26°

A

D

Figure 4.7

take note

• BC 85, 10°, B 64°

Having found AC, we can now find the height of the building. h AC h 苷 AC sin 16 85 sin 64 苷 sin 16 ⬇ 121 feet sin 10

In Example 3 we rounded the height

sin 16 苷

of the building to two significant digits to comply with the rounding conventions given on page 140.

• Substitute for AC.

The height of the building, to two significant digits, is 120 feet.

»

Try Exercise 30, page 299

4.1

297

The Law of Sines

In navigation and surveying problems, there are two commonly used methods for specifying direction. The angular direction in which a craft is pointed is called the heading. Heading is expressed in terms of an angle measured clockwise from north. Figure 4.8 shows a heading of 65° and a heading of 285°. The angular direction used to locate one object in relation to another object is called the bearing. Bearing is expressed in terms of the acute angle formed by a north–south line and the line of direction. Figure 4.9 shows a bearing of N38°W and a bearing of S15°E. N N N 38° W W

E

65°

S 15° E S

285°

Figure 4.8 QUESTION

Can a bearing of N50°E be written as N310°W?

EXAMPLE 4 N

Figure 4.9

»

Solve an Application

A ship with a heading of 330° first sighted a lighthouse at a bearing of N65°E. After traveling 8.5 miles, the ship observed the lighthouse at a bearing of S50°E. Find the distance from the ship to the lighthouse when the first sighting was made.

C 50° 30°

Solution

N

8.5 mi

Lighthouse 30° 65°

B c

330°

Use the given information to draw a diagram. See Figure 4.10 which shows that the measure of ⬔CAB 苷 65 30 苷 95 , the measure of ⬔BCA 苷 50 30 苷 20 , and B 苷 180 95 20 苷 65 . Use triangle ABC and the Law of Sines to find c. b c 苷 sin B sin C 8.5 c 苷 sin 65 sin 20 8.5 sin 20 c苷 ⬇ 3.2 sin 65

A

Starting point

Figure 4.10

• b 8.5, B 65°, C 20°

The lighthouse was 3.2 miles (to two significant digits) from the ship when the first sighting was made.

»

Try Exercise 40, page 300

ANSWER

No. A bearing is always expressed using an acute angle.

298

Chapter 4

Applications of Trigonometry

Topics for Discussion 1. Is it possible to solve a triangle if the only given information consists of the measures of the three angles of the triangle? Explain. 2. Explain why it is not possible (in general) to use the Law of Sines to solve a triangle for which we are given only the lengths of all the sides. 3. Draw a triangle with dimensions A 苷 30 , c 苷 3 inches, and a 苷 2.5 inches. Is your answer unique? That is, can more than one triangle with the given dimensions be drawn?

Exercise Set 4.1 In Exercises 1 to 44, round answers according to the rounding conventions on page 140. In Exercises 1 to 14, solve the triangles.

17. C 苷 65 , b 苷 10, c 苷 8.0

18. A 苷 42 , a 苷 12, c 苷 18

19. A 苷 30 , a 苷 1.0, b 苷 2.4

1. A 苷 42 , B 苷 61 , a 苷 12

20. B 苷 22.6 , b 苷 5.55, a 苷 13.8

2. B 苷 25 , C 苷 125 , b 苷 5.0

21. A 苷 14.8 , c 苷 6.35, a 苷 4.80

3. A 苷 110 , C 苷 32 , b 苷 12

22. C 苷 37.9 , b 苷 3.50, c 苷 2.84

4. B 苷 28 , C 苷 78 , c 苷 44

23. C 苷 47.2 , a 苷 8.25, c 苷 5.80

5. A 苷 132 , a 苷 22, b 苷 16

24. B 苷 52.7 , b 苷 12.3, c 苷 16.3

6. B 苷 82.0 , b 苷 6.0, c 苷 3.0

25. B 苷 117.32 , b 苷 67.25, a 苷 15.05

7. A 苷 22.5 , B 苷 112.4 , a 苷 16.3

26. A 苷 49.22 , a 苷 16.92, c 苷 24.62

8. A 苷 21.5 , B 苷 104.2 , c 苷 57.4

27. A 苷 20.5 , a 苷 10.3, c 苷 14.1

9. A 苷 82.0 , B 苷 65.4 , b 苷 36.5

28. B 苷 41.2 , a 苷 31.5, b 苷 21.6

11. A 苷 33.8 , C 苷 98.5 , c 苷 102 12. B 苷 36.9 , C 苷 69.2 , a 苷 166 13. C 苷 114.2 , c 苷 87.2, b 苷 12.1

Elizabeth City Nags Head

North Carolina

64.9°

In Exercises 15 to 28, solve the triangles that exist.

15. A 苷 37 , c 苷 40, a 苷 28 16. B 苷 32 , c 苷 14, b 苷 9.0

New Bern

10

5m

14. A 苷 54.32 , a 苷 24.42, c 苷 16.92

29. HURRICANE WATCH A satellite weather map shows a hurricane off the coast of North Carolina. Use the information in the map to find the distance from the hurricane to Nags Head.

ile s

10. B 苷 54.8 , C 苷 72.6 , a 苷 14.4

39.4° Morehead City

Hurricane

4.1 30. NAVAL MANEUVERS The distance between an aircraft carrier and a Navy destroyer is 7620 feet. The angle of elevation from the destroyer to a helicopter is 77.2°, and the angle of elevation from the aircraft carrier to the helicopter is 59.0°. The helicopter is in the same vertical plane as the two ships, as shown in the following figure. Use this data to determine the distance x from the helicopter to the aircraft carrier.

The distance AC from the tee directly to the pin is 365 yards. Angle A measures 11.2° and angle C measures 22.9°. a. Find the distance AB that the golfer drove the ball. b. Find the distance BC from the present position of the ball to the pin.

Helicopter

C

x

59.0° Aircraft carrier

299

The Law of Sines

B 77.2°

7620 feet

Navy destroyer

31. CHOOSING A GOLF STRATEGY The following diagram shows two ways to play a golf hole. One is to hit the ball down the fairway on your first shot and then hit an approach shot to the green on your second shot. A second way is to hit directly toward the pin. Due to the water hazard, this is a more risky strategy. The distance AB is 165 yards, BC is 155 yards, and angle A 苷 42.0 . Find the distance AC from the tee directly to the pin. Assume that angle B is an obtuse angle.

C

A

33. DISTANCE TO A HOT AIR BALLOON The angle of elevation to a balloon from one observer is 67°, and the angle of elevation from another observer, 220 feet away, is 31°. If the balloon is in the same vertical plane as the two observers and between them, find the distance of the balloon from the first observer. 34. HEIGHT OF A SPACE SHUTTLE Use the Law of Sines to solve Example 6 on page 141. Compare this method with the method used in Example 6. Which method do you prefer? Explain. 35. RUNWAY REPLACEMENT An airport runway is 3550 feet long and has an incline of 3.0°. The airport planning committee plans to replace this runway with a new runway, as shown in the following figure. The new runway will be inclined at an angle of 2.2°. What will be the length of the new runway?

B

A C

New runway 2.2° A

32. DRIVING DISTANCE A golfer drives a golf ball from the tee at point A to point B, as shown in the following diagram.

Current runway B

3.0° (Not drawn to scale)

Horizontal

300

Chapter 4

Applications of Trigonometry

36. HEIGHT OF A KITE Two observers, in the same vertical plane as a kite and at a distance of 30 feet apart, observe the kite at angles of 62° and 78°, as shown in the following diagram. Find the height of the kite.

39. HEIGHT OF A HILL A surveying team determines the height of a hill by placing a 12-foot pole at the top of the hill and measuring the angles of elevation to the bottom and the top of the pole. They find the angles of elevation to be as shown in the following figure. Find the height of the hill.

12 ft

75° 70° d d

62°

78°

30 ft

37. LENGTH OF A GUY WIRE A telephone pole 35 feet high is situated on an 11° slope from the horizontal. The measure of angle CAB is 21°. Find the length of the guy wire AC.

A

21°

40. DISTANCE TO A FIRE Two fire lookouts are located on mountains 20 miles apart. Lookout B is at a bearing of S65°E from lookout A. A fire was sighted at a bearing of N50°E from A and at a bearing of N8°E from B. Find the distance of the fire from lookout A.

C

41. DISTANCE TO A LIGHTHOUSE A navigator on a ship sights a lighthouse at a bearing of N36°E. After traveling 8.0 miles at a heading of 332°, the ship sights the lighthouse at a bearing of S82°E. How far is the ship from the lighthouse at the second sighting?

B

42. MINIMUM DISTANCE The navigator on a ship traveling due east at 8 mph sights a lighthouse at a bearing of S55°E. One hour later the lighthouse is sighted at a bearing of S25°W. Find the closest the ship came to the lighthouse.

11°

38. DIMENSIONS OF A PLOT OF LAND Three roads intersect in such a way as to form a triangular piece of land. See the accompanying figure. Find the lengths of the other two sides of the land.

43. DISTANCE BETWEEN AIRPORTS An airplane flew 450 miles at a bearing of N65°E from airport A to airport B. The plane then flew at a bearing of S38°E to airport C. Find the distance from A to C if the bearing from airport A to airport C is S60°E. 44. LENGTH OF A BRACE A 12-foot solar panel is to be installed on a roof with a 15° pitch. Find the length of the vertical brace d if the panel must be installed to make a 40° angle with the horizontal.

320 ft 54°

47°

12 ft d

15°

4.1

The Law of Sines

301

Connecting Concepts 45. DISTANCES BETWEEN HOUSES House B is located at a bearing of N67°E from house A. House C is 300 meters from house A at a bearing of S68°E. House B is located at a bearing of N11°W from house C. Find the distance from house A to house B.

46. Show that for any triangle ABC,

a b sin A sin B . 苷 b sin B

47. Show that for any triangle ABC,

a b sin A sin B . 苷 b sin B

48. Show that for any triangle ABC,

a b sin A sin B . 苷 a b sin A sin B

49.

MAXIMUM LENGTH OF A ROD The longest rod that can be carried horizontally around a corner from a hall 3 meters wide into one that is 5 meters wide is the

Projects 1. FERMAT’S PRINCIPLE AND SNELL’S LAW State Fermat’s Principle and Snell’s Law. The refractive index of the glass in a ring is found to be 1.82. The refractive index of a particular diamond in a ring is 2.38. Use Snell’s Law to explain what this means in terms of the reflective properties of the glass and the diamond.

minimum of the length L of the black dashed line shown in the figure below. 3m

3m 5m

5m

θ

Use similar triangles to show that the length L is a function of the angle , given by L共兲苷

5 3 sin cos

Graph L and estimate the minimum value of L. Round to the nearest hundredth of a meter.

302

Chapter 4

Applications of Trigonometry

The Law of Cosines and Area

Section 4.2 ■ ■ ■

P R E PA R E F O R T H I S S E C T I O N

The Law of Cosines Area of a Triangle Heron’s Formula

Prepare for this section by completing the following exercises. The answers can be found on page A20.

PS1. Evaluate 兹a2 b2 2ab cos C for a 苷 10.0, b 苷 15.0, and C 苷 110.0 . Round your result to the nearest tenth. [2.3] PS2. Find the area of a triangle with a base of 6 inches and a height of 8.5 inches. PS3. Solve c2 苷 a2 b2 2ab cos C for C. [3.5] PS4. The semiperimeter of a triangle is defined as one-half the perimeter of the triangle. Find the semiperimeter of a triangle with sides of 6 meters, 9 meters, and 10 meters. PS5. Evaluate 兹s共s a兲共s b兲共s c兲 for a 苷 3, b 苷 4, c 苷 5, and abc s苷 . 2 PS6. State a relationship between the lengths a, b, and c in the triangle shown at the right. [1.2]

c

b

a

The Law of Cosines The Law of Cosines can be used to solve triangles in which two sides and the included angle (SAS) are known or in which three sides (SSS) are known. Consider the triangle in Figure 4.11. The height BD is drawn from B perpendicular to the x-axis. The triangle BDA is a right triangle, and the coordinates of B are 共a cos C, a sin C兲. The coordinates of A are 共b, 0兲. Using the distance formula, we can find the distance c.

y B(a cos C, a sin C)

c

a C D

(0, 0)

A(b, 0) b

Figure 4.11

x

c 苷兹共a cos C b兲2 共a sin C 0兲2 c 2 苷 a2 cos2 C 2ab cos C b 2 a2 sin2 C c 2 苷 a2共cos2 C sin2 C兲 b 2 2ab cos C c 2 苷 a2 b2 2ab cos C

The Law of Cosines If A, B, and C are the measures of the angles of a triangle and a, b, and c are the lengths of the sides opposite these angles, then c2 苷 a2 b 2 2ab cos C a2 苷 b 2 c2 2bc cos A b 2 苷 a2 c2 2ac cos B

4.2

EXAMPLE 1

»

303

The Law of Cosines and Area

Use the Law of Cosines (SAS)

In triangle ABC, B 苷 110.0 , a 苷 10.0 centimeters, and c 苷 15.0 centimeters. See Figure 4.12. Find b. A

Solution The Law of Cosines can be used because two sides and the included angle are known. b

c = 15.0 cm

b 2 苷 a2 c 2 2ac cos B 苷 10.0 2 15.0 2 2共10.0兲共15.0兲 cos 110.0 b 苷兹10.0 2 15.0 2 2共10.0兲共15.0兲 cos 110.0 b ⬇ 20.7 centimeters

110.0° B

a = 10.0 cm

Figure 4.12

C

»

Try Exercise 12, page 308

In the next example we know the length of each side, but we do not know the measure of any of the angles. EXAMPLE 2

»

Use the Law of Cosines (SSS)

In triangle ABC, a 苷 32 feet, b 苷 20 feet, and c 苷 40 feet. Find B. This is the SSS case.

Solution b 2 苷 a2 c 2 2ac cos B cos B 苷苷

a2 c 2 b 2 2ac

• Solve for cos B.

322 40 2 20 2 2共32兲共40兲

• Substitute for a, b, and c.

冉

B 苷 cos1

冊

322 40 2 20 2 2共32兲共40兲

B ⬇ 30

• Solve for angle B. • To the nearest degree

»

Try Exercise 18, page 308

EXAMPLE 3

»

Solve an Application Using the Law of Cosines

A boat sailed 3.0 miles at a heading of 78° and then turned to a heading of 138° and sailed another 4.3 miles. Find the distance and the bearing of the boat from the starting point. Continued

䉴

304

Chapter 4

Applications of Trigonometry Solution

N N 78° 3.0 mi

B

Sketch a diagram (see Figure 4.13). First find the measure of angle B in triangle ABC.

138°

B 苷 78 共180 138 兲苷 120

78°

A

α

4.3 mi

Use the Law of Cosines first to find b and then to find A.

b

b 2 苷 a2 c2 2ac cos B 苷 4.32 3.0 2 2共4.3兲共3.0兲 cos 120 b 苷兹4.32 3.02 2共4.3兲共3.0兲 cos 120 b ⬇ 6.4 miles

C

Figure 4.13

cos A 苷

b 2 c 2 a2 2bc

A 苷 cos1

冉

冊

• Substitute for a, c, and B.

冉

• Solve the Law of Cosines for cos A.

冊

b 2 c 2 a2 6.42 3.0 2 4.32 ⬇ cos1 ⬇ 35° 2bc 共2兲共6.4兲共3.0兲

The bearing of the present position of the boat from the starting point A can be determined by calculating the measure of angle in Figure 4.13.

take note The measure of angle A in Example 3 can also be determined by using the Law of Sines.

⬇ 180 共78 35 兲苷 67 The distance is approximately 6.4 miles, and the bearing (to the nearest degree) is S67°E.

»

Try Exercise 52, page 310

There are five different cases that we may encounter when solving an oblique triangle. Each case is listed below under the law that can be used to solve the triangle. Choosing Between the Law of Sines and the Law of Cosines Apply the Law of Sines to solve an oblique triangle for each of the following cases. ASA The measures of two angles of the triangle and the length of the included side are known. AAS The measures of two angles of the triangle and the length of a side opposite one of these angles are known. SSA The lengths of two sides of the triangle and the measure of an angle opposite one of these sides are known. This case is called the ambiguous case. It may yield one solution, two solutions, or no solution. Apply the Law of Cosines to solve an oblique triangle for each of the following cases. SSS The lengths of all three sides of the triangle are known. After finding the measure of an angle, you can complete your solution by using the Law of Sines. SAS The lengths of two sides of the triangle and the measure of the included angle are known. After finding the measure of the third side, you can complete your solution by using the Law of Sines.

4.2 B

a

QUESTION

b

In triangle ABC, A 苷 40 , C 苷 60 , and b 苷 114. Should you use the Law of Sines or the Law of Cosines to solve this triangle?

Area of a Triangle

A

Acute triangle

1 bh can be used to find the area of a triangle when the base and 2 height are given. In this section we will find the areas of triangles when the height is not given. We will use K for the area of a triangle because the letter A is often used to represent the measure of an angle. Consider the areas of the acute and obtuse triangles in Figure 4.14. The formula A 苷

B

h

305

c

h

C

The Law of Cosines and Area

c a C

b

A

Height of each triangle:

Obtuse triangle

Figure 4.14

Area of each triangle:

h 苷 c sin A 1 K 苷 bh 2 1 K 苷 bc sin A 2

• Substitute for h.

Thus we have established the following theorem.

Area of a Triangle

take note Because each formula requires

The area K of triangle ABC is one-half the product of the lengths of any two sides and the sine of the included angle. Thus

two sides and the included angle, it is necessary to learn only one

K苷

formula.

EXAMPLE 4 B

1 bc sin A 2

»

K苷

1 ab sin C 2

K苷

1 ac sin B 2

Find the Area of a Triangle

Given angle A 苷 62 , b 苷 12 meters, and c 苷 5.0 meters, find the area of triangle ABC.

c = 5.0 62° A

b = 12

Figure 4.15

C

Solution In Figure 4.15, two sides and the included angle of the triangle are given. Using the formula for area, we have K苷

1 1 bc sin A 苷共12兲共5.0兲共sin 62 兲 ⬇ 26 square meters 2 2

»

Try Exercise 30, page 309

ANSWER

Because the measure of two angles and the length of the included side are given, the triangle can be solved by using the Law of Sines.

306

Chapter 4

Applications of Trigonometry When two angles and an included side are given, the Law of Sines is used to derive a formula for the area of a triangle. First, solve for c in the Law of Sines. c b 苷 sin C sin B b sin C c苷 sin B Substitute for c in the formula K 苷

1 bc sin A. 2

冉冊

1 1 b sin C bc sin A 苷 b sin A 2 2 sin B b 2 sin C sin A K苷 2 sin B K苷

In like manner, the following two alternative formulas can be derived for the area of a triangle.

EXAMPLE 5

K苷

a2 sin B sin C 2 sin A

»

Find the Area of a Triangle

and K 苷

c2 sin A sin B 2 sin C

Given A 苷 32 , C 苷 77 , and a 苷 14 inches, find the area of triangle ABC.

Solution To use the above area formula, we need to know two angles and the included side. Therefore, we need to determine the measure of angle B. B 苷 180 32 77 苷 71 Thus K苷

a2 sin B sin C 142 sin 71 sin 77 苷 ⬇ 170 square inches 2 sin A 2 sin 32

»

Try Exercise 32, page 309

Math Matters Recent findings indicate that Heron’s formula for finding the area of a triangle was first discovered by Archimedes. However, the formula is called Heron’s formula in honor of the geometer Heron of Alexandria (A.D. 50), who gave an ingenious proof of the theorem in his work Metrica. Because Heron of Alexandria was also known as Hero, some texts refer to Heron’s formula as Hero’s formula.

Heron’s Formula The Law of Cosines can be used to derive Heron’s formula for the area of a triangle in which three sides of the triangle are given. Heron’s Formula for Finding the Area of a Triangle If a, b, and c are the lengths of the sides of a triangle, then the area K of the triangle is K 苷兹s共s a兲共s b兲共s c兲,

where s 苷

1 共a b c兲 2

4.2

The Law of Cosines and Area

307

Because s is one-half the perimeter of the triangle, it is called the semiperimeter.

EXAMPLE 6

»

Find an Area by Heron’s Formula

Find, to two significant digits, the area of the triangle with a 苷 7.0 meters, b 苷 15 meters, and c 苷 12 meters.

Solution Calculate the semiperimeter s. s苷

a b c 7.0 15 12 苷苷 17 2 2

Use Heron’s formula. K 苷兹s共s a兲共s b兲共s c兲苷兹17共17 7.0兲共17 15兲共17 12兲苷兹1700 ⬇ 41 square meters

»

Try Exercise 40, page 309

EXAMPLE 7

»

Use Heron’s Formula to Solve an Application

The original portion of the Luxor Hotel in Las Vegas has the shape of a square pyramid. Each face of the pyramid is an isosceles triangle with a base of 646 feet and sides of length 576 feet. Assuming that the glass on the exterior of the Luxor Hotel costs $35 per square foot, determine the cost of the glass, to the nearest $10,000, for one of the triangular faces of the hotel.

Solution The lengths (in feet) of the sides of a triangular face are a 苷 646, b 苷 576, and c 苷 576. s苷

a b c 646 576 576 苷苷 899 feet 2 2

K 苷兹s共s a兲共s b兲共s c兲苷兹899共899 646兲共899 576兲共899 576兲苷兹23,729,318,063 ⬇ 154,043 square feet The cost C of the glass is the product of the cost per square foot and the area. C ⬇ 35 154,043 苷 5,391,505 The pyramid portion of the Luxor Hotel in Las Vegas, Nevada

The approximate cost of the glass for one face of the Luxor Hotel is $5,390,000.

»

Try Exercise 60, page 310

308

Chapter 4

Applications of Trigonometry

Topics for Discussion 1. Explain why there is no triangle that has sides of lengths a 苷 2 inches, b 苷 11 inches, and c 苷 3 inches. 2. The Pythagorean Theorem is a special case of the Law of Cosines. Explain. 3. To solve a triangle in which the lengths of the three sides are given (SSS), a mathematics professor recommends the following procedure. ii(i) Use the Law of Cosines to find the measure of the largest angle. i(ii) Use the Law of Sines to find the measure of a second angle. (iii) Find the measure of the third angle by using the formula A B C 苷 180 . Explain why this procedure is easier than using the Law of Cosines to find the measure of all three angles. 4. Explain why the Law of Cosines cannot be used to solve a triangle in which you are given the measures of two angles and the length of the included side (ASA).

Exercise Set 4.2 In Exercises 1 to 52, round answers according to the rounding conventions on page 140. In Exercises 1 to 14, find the third side of the triangle.

13. a 苷 122, c 苷 55.9, B 苷 44.2 14. b 苷 444.8, c 苷 389.6, A 苷 78.44

1. a 苷 12, b 苷 18, C 苷 44 2. b 苷 30, c 苷 24, A 苷 120

In Exercises 15 to 24, given three sides of a triangle, find the specified angle.

3. a 苷 120, c 苷 180, B 苷 56

15. a 苷 25, b 苷 32, c 苷 40; find A.

4. a 苷 400, b 苷 620, C 苷 116

16. a 苷 60, b 苷 88, c 苷 120; find B.

5. b 苷 60, c 苷 84, A 苷 13

17. a 苷 8.0, b 苷 9.0, c 苷 12; find C.

6. a 苷 122, c 苷 144, B 苷 48

18. a 苷 108, b 苷 132, c 苷 160; find A.

7. a 苷 9.0, b 苷 7.0, C 苷 72

19. a 苷 80.0, b 苷 92.0, c 苷 124; find B.

8. b 苷 12, c 苷 22, A 苷 55

20. a 苷 166, b 苷 124, c 苷 139; find B.

9. a 苷 4.6, b 苷 7.2, C 苷 124

21. a 苷 1025, b 苷 625.0, c 苷 1420; find C.

10. b 苷 12.3, c 苷 14.5, A 苷 6.5

22. a 苷 4.7, b 苷 3.2, c 苷 5.9; find A.

11. a 苷 25.9, c 苷 33.4, B 苷 84.0

23. a 苷 32.5, b 苷 40.1, c 苷 29.6; find B.

12. a 苷 14.2, b 苷 9.30, C 苷 9.20

24. a 苷 112.4, b 苷 96.80, c 苷 129.2; find C.

4.2

309

The Law of Cosines and Area

In Exercises 25 to 28, solve the triangle.

25. A 苷 39.4 , b 苷 15.5, c 苷 17.2 26. C 苷 98.4 , a 苷 141, b 苷 92.3 27. a 苷 83.6, b 苷 144, c 苷 98.1

t

26 ft

90

28. a 苷 25.4, b 苷 36.3, c 苷 38.2 In Exercises 29 to 40, find the area of the given triangle. Round each area to the same number of significant digits given for each of the given sides.

fee

30. B 苷 127 , a 苷 32, c 苷 25

B-2 BOMBER The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet in length. The angle between the wing’s leading edges 共⬔ABC兲 is 109.05°. What is the wing span (the distance from A to C) of the B-2 Bomber?

31. A 苷 42 , B 苷 76 , c 苷 12

B

29. A 苷 105 , b 苷 12, c 苷 24

44.

32. B 苷 102 , C 苷 27 , a 苷 8.5 33. a 苷 16, b 苷 12, c 苷 14

C

A B-2 Stealth Bomber

34. a 苷 32, b 苷 24, c 苷 36 35. B 苷 54.3 , a 苷 22.4, b 苷 26.9 36. C 苷 18.2 , b 苷 13.4, a 苷 9.84 37. A 苷 116 , B 苷 34 , c 苷 8.5

45. ANGLE BETWEEN THE DIAGONALS OF A BOX The rectangular box in the figure measures 6.50 feet by 3.25 feet by 4.75 feet. Find the measure of the angle u that is formed by the union of the diagonal shown on the front of the box and the diagonal shown on the right side of the box.

38. B 苷 42.8 , C 苷 76.3 , c 苷 17.9 39. a 苷 3.6, b 苷 4.2, c 苷 4.8

4.75 ft

40. a 苷 10.2, b 苷 13.3, c 苷 15.4 41. DISTANCE BETWEEN AIRPORTS A plane leaves airport A and travels 560 miles to airport B at a bearing of N32°E. The plane leaves airport B and travels to airport C 320 miles away at a bearing of S72°E. Find the distance from airport A to airport C. 42. LENGTH OF A STREET A developer owns a triangular lot at the intersection of two streets. The streets meet at an angle of 72°, and the lot has 300 feet of frontage along one street and 416 feet of frontage along the other street. Find the length of the third side of the lot. 43. BASEBALL In a baseball game, a batter hits a ground ball 26 feet in the direction of the pitcher’s mound. See the figure at the top of the next column. The pitcher runs forward and reaches for the ball. At that moment, how far is the ball from first base? (Note: A baseball infield is a square that measures 90 feet on each side.)

θ 3.25 ft 6.50 ft

46. SUBMARINE RESCUE MISSION Use the distances shown in the following figure to determine the depth of the submarine below the surface of the water. Assume that the line segment between the surface ships is directly above the submarine. 615 feet 499 feet

Submarine

629 feet

310

Chapter 4

Applications of Trigonometry

47. DISTANCE BETWEEN SHIPS Two ships left a port at the same time. One ship traveled at a speed of 18 mph at a heading of 318°. The other ship traveled at a speed of 22 mph at a heading of 198°. Find the distance between the two ships after 10 hours of travel.

Piston C 4 cm

16 cm c

A

B

48. DISTANCE ACROSS A LAKE Find the distance across the lake shown in the figure. a. Use the Law of Cosines to find an equation that relates c and A. A

b. Use the quadratic formula to solve the equation in a. for c.

136 m

c. Use the equation from b. to find c when A 苷 55 . Round to the nearest centimeter. B

78.0°

d. Use the Law of Sines to find c when A 苷 55 . Round to the nearest centimeter. How does this result compare with the result in c.?

162 m

C

55. AREA OF A TRIANGULAR LOT Find the area of a triangular piece of land that is bounded by sides of 236 meters, 620 meters, and 814 meters. Round to the nearest hundred square meters.

49. GEOMETRY A regular hexagon is inscribed in a circle with a radius of exactly 40 centimeters. Find the exact length of one side of the hexagon.

56. GEOMETRY Find the exact area of a parallelogram with sides of exactly 8 feet and 12 feet. The shorter diagonal is exactly 10 feet.

50. ANGLE BETWEEN BOUNDARIES OF A LOT A triangular city lot has sides of 224 feet, 182 feet, and 165 feet. Find the angle between the longer two sides of the lot.

57. GEOMETRY Find the exact area of a square inscribed in a circle with a radius of exactly 9 inches.

51. DISTANCE TO A PLANE A plane traveling at 180 mph passes 400 feet directly over an observer. The plane is traveling along a straight path with an angle of elevation of 14°. Find the distance of the plane from the observer 10 seconds after the plane has passed directly overhead. 52. DISTANCE BETWEEN SHIPS A ship leaves a port at a speed of 16 mph at a heading of 32°. One hour later another ship leaves the port at a speed of 22 mph at a heading of 254°. Find the distance between the ships 4 hours after the first ship leaves the port. 53. DISTANCE AND BEARING FROM A STARTING POINT A plane flew 181 miles at a heading of 108.5° and then turned to a heading of 124.6° and flew another 225 miles. Find the distance and the bearing of the plane from the starting point. 54. ENGINE DESIGN An engine has a 16-centimeter connecting rod that is attached to a rotating crank with a 4-centimeter radius. See the following figure.

58. GEOMETRY Find the exact area of a regular hexagon inscribed in a circle with a radius of exactly 24 centimeters. 59. COST OF A LOT A commercial piece of real estate is priced at $2.20 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 212 feet by 185 feet by 240 feet. 60. COST OF A LOT An industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet. 61. AREA OF A PASTURE Find the number of acres in a pasture whose shape is a triangle measuring 800 feet by 1020 feet by 680 feet. Round to the nearest hundredth of an acre. (An acre is 43,560 square feet.) 62. AREA OF A HOUSING TRACT Find the number of acres in a housing tract whose shape is a triangle measuring 420 yards by 540 yards by 500 yards. Round to the nearest hundredth of an acre. (An acre is 4840 square yards.)

4.2 The identity at the right is one of Mollweide’s formulas. It applies to any triangle ABC.

ab c

冉冊冉冊

An architect has determined the following dimensions for 䉭ABC and 䉭DEF:

AB 2 C cos 2

sin

䉭ABC:

The formula is intriguing because it contains the dimensions of all angles and all sides of triangle ABC. The formula can be used to check whether a triangle has been solved correctly. Substitute the dimensions of a given triangle in the formula and compare the value of the left side of the formula with the value of the right side. If the two results are not reasonably close, then you know that at least one dimension is incorrect. The results generally will not be identical because each dimension is an approximation. In Exercises 63 and 64, you can assume that a triangle has an incorrect dimension if the value of the left side and the value of the right side of the above formula differ by more than 0.02.

A 苷 53.5 , B 苷 86.5 , C 苷 40.0 a 苷 13.0 feet, b 苷 16.1 feet, c 苷 10.4 feet

䉭DEF: D 苷 52.1 , E 苷 59.9 , F 苷 68.0 d 苷 17.2 feet, e 苷 21.3 feet, f 苷 22.8 feet Use Mollweide’s formula to determine if either 䉭ABC or 䉭DEF has an incorrect dimension. 64.

CHECK DIMENSIONS OF TRUSSES The following diagram shows some of the steel trusses in a railroad bridge. F D C

A

63.

CHECK DIMENSIONS OF TRUSSES The following diagram shows some of the steel trusses in an airplane hangar.

B

E

F

D

C

A

311

The Law of Cosines and Area

A structural engineer has determined the following dimensions for 䉭ABC and 䉭DEF:

B

E

䉭ABC:

A 苷 34.1 , B 苷 66.2 , C 苷 79.7 a 苷 9.23 feet, b 苷 15.1 feet, c 苷 16.2 feet

䉭DEF: D 苷 45.0 , E 苷 56.2 , F 苷 78.8 d 苷 13.6 feet, e 苷 16.0 feet, f 苷 18.9 feet Use Mollweide’s formula to determine if either 䉭ABC or 䉭DEF has an incorrect dimension.

Connecting Concepts 65. Find the measure of the angle formed by the sides P1 P2 and P1 P3 of a triangle with vertices at P1共2, 4兲, P2共2, 1兲, and P3共4, 3兲.

69. Find the volume of the triangular prism shown in the figure.

66. Given a triangle ABC, prove that for any triangle ABC

72°

4 in.

4 in.

18 in.

a 苷 b c 2bc cos A 2

2

2

67. Use the Law of Cosines to show that for any triangle ABC cos A 苷

共b c a兲共b c a兲 1 2bc

70. Show that the area of the circumscribed triangle in the folabc lowing figure is K 苷 rs, where s 苷 . 2 B

68. Prove that K 苷 xy sin A for a parallelogram, where x and y are the lengths of adjacent sides, A is the measure of the angle between side x and side y, and K is the area of the parallelogram.

c

a r

A

b

C

312

Chapter 4

Applications of Trigonometry

Projects 1. VISUAL INSIGHT Give the reason that justifies each step in the following proof of the Law of Cosines. bd 苷共a c兲e d

bd 苷共a c兲共a c兲

e

b共2a cos C b兲苷共a c兲共a c兲

C

c2 苷 a2 b2 2ab cos C

a

a Center of circle

a

Section 4.3 ■ ■ ■ ■ ■ ■

■

Vectors Unit Vectors Applications of Vectors Dot Product Scalar Projection Parallel and Perpendicular Vectors Work: An Application of the Dot Product

b

c

2ab cos C b2 苷 a2 c2

Vectors P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A20.

PS1. Evaluate:

冑冉冊冉冊 3 5

2

4 5

2

PS2. Use a calculator to evaluate 10 cos 228 . Round to the nearest thousandth. [2.3] PS3. Solve tan a 苷

兩

兩

兹3 for , where is an acute angle measured in degrees. [3.5] 3

17

for , where is an obtuse angle measured in degrees. Round to 兹338 the nearest tenth of a degree. [3.5]

PS4. Solve cos a 苷

PS5. Rationalize the denominator of

PS6. Rationalize the denominator of

1 兹5

.

28 兹68

.

Vectors In scientific applications some measurements, such as area, mass, distance, speed, and time, are completely described by a real number and a unit. Examples include 30 square feet (area), 25 meters兾second (speed), and 5 hours (time). These measurements are called scalar quantities, and the number used to indicate the magnitude of the measurement is called a scalar. Two other examples of scalar quantities are volume and temperature. For other quantities, in addition to the numerical and unit description, it is also necessary to include a direction to describe the quantity completely. For

4.3

B 25 lb 25 lb

C

Figure 4.16

Terminal point

Definition of a Vector

B

A vector is a directed line segment. The length of the line segment is the magnitude of the vector, and the direction of the vector is measured by an angle.

A Initial point

Figure 4.17

Figure 4.18

v

2v

313

example, applying a force of 25 pounds at various angles to a small metal box will influence how the box moves. In Figure 4.16, applying the 25-pound force straight down (A) will not move the box to the left. However, applying the 25-pound force parallel to the floor (C) will move the box along the floor. Vector quantities have a magnitude (numerical and unit description) and a direction. Force is a vector quantity. Velocity is another. Velocity includes the speed (magnitude) and a direction. A velocity of 40 mph east is different from a velocity of 40 mph north. Displacement is another vector quantity; it consists of distance (a scalar) moved in a certain direction. For example, we might speak of a displacement of 13 centimeters at an angle of 15° from the positive x-axis.

A 25 lb

Vectors

1 v 2

v

v

Figure 4.19

50 m

30 m

37° 40 m

Figure 4.20

U+V

nt

ulta Res

V

−1.5v

The point A for the vector in Figure 4.17 is called the initial point (or tail) of the vector, and the point B is the terminal point (or head) of the vector. An arrow l l over the letters 共AB兲, an arrow over a single letter 共V 兲, or boldface type (AB or V) is used to denote a vector. The magnitude of the vector is the length of the line segl l ment and is denoted by 储 AB 储 , 储 V 储 , 储 AB 储 , or 储 V 储 . Equivalent vectors have the same magnitude and the same direction. The vectors in Figure 4.18 are equivalent. They have the same magnitude and direction. Multiplying a vector by a positive real number (other than 1) changes the magnitude of the vector but not its direction. If v is any vector, then 2v is the vector that has the same direction as v but is twice the magnitude of v. The multiplication of 2 and v is called the scalar multiplication of the vector v and the scalar 2. Multiplying a vector by a negative number a reverses the direction of the vector and multiplies the magnitude of the vector by 兩a兩. See Figure 4.19. The sum of two vectors, called the resultant vector or the resultant, is the single equivalent vector that will have the same effect as the application of those two vectors. For example, a displacement of 40 meters along the positive x-axis and then 30 meters in the positive y direction is equivalent to a vector of magnitude 50 meters at an angle of approximately 37° to the positive x-axis. See Figure 4.20. Vectors can be added graphically by using the triangle method or the parallelogram method. In the triangle method, shown in Figure 4.21, the initial point of V is placed at the terminal point of U. The vector connecting the initial point of U with the terminal point of V is the sum U V. The parallelogram method of adding two vectors graphically places the initial point of the two vectors U and V together, as in Figure 4.22. Complete the parallelogram so that U and V are sides of the parallelogram. The diagonal beginning at the initial point of the two vectors is U V. To find the difference between two vectors, first rewrite the expression as V U 苷 V 共U兲. The difference is shown geometrically in Figure 4.23.

U

Figure 4.21

V

U+V

V−U

t

ltan

u Res

U

Figure 4.22

V

−U

U

Figure 4.23

314

Chapter 4

Applications of Trigonometry By introducing a coordinate plane, it is possible to develop an analytic approach to vectors. Recall from our discussion of equivalent vectors that a vector can be moved in the plane as long as the magnitude and direction are not changed. With this in mind, consider AB, whose initial point is A共2, 1兲 and whose terminal point is B共3, 4兲. If this vector is moved so that the initial point is at the origin O, the terminal point becomes P共5, 5兲, as shown in Figure 4.24. The vector OP is equivalent to the vector AB. In Figure 4.25, let P1共x1 , y1兲 be the initial point of a vector and P2共x2 , y2兲 its terminal point. Then an equivalent vector OP has its initial point at the origin and its terminal point at P共a, b兲, where a 苷 x2 x1 and b 苷 y2 y1 . The vector OP can be denoted by v 苷具a, b典; a and b are called the components of the vector. y 5

P(−5, 5)

y

P(a, b)

B(−3, 4) P2(x 2 , y2)

x

O −5

x

O A(2, −1)

P1(x 1, y1)

Figure 4.24

EXAMPLE 1

»

Figure 4.25

Find the Components of a Vector

Find the components of the vector AB whose initial point is A共2, 1兲 and whose terminal point is B共2, 6兲. Determine a vector v that is equivalent to AB and has its initial point at the origin.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

The components of AB are 具a, b典, where a 苷 x2 x1 苷 2 2 苷 4

y

P(−4, 7)

and b 苷 y2 y1 苷 6 共1兲苷 7

7 B(−2, 6)

Thus v 苷具4, 7典. v AB

x

O

−5

A(2, −1)

Figure 4.26

»

Try Exercise 10, page 325

The magnitude and direction of a vector can be found from its components. For instance, the terminal point of vector v sketched in Figure 4.26 is the ordered pair 共4, 7兲. Applying the Pythagorean Theorem, we find 储 v 储苷兹共4兲2 72 苷兹16 49 苷兹65

4.3

Vectors

315

Let be the angle made by the positive x-axis and v. Let be the reference angle for . Then

兩兩兩兩

b 7 7 苷苷 a 4 4 7 苷 tan1 ⬇ 60 4 苷 180 60 苷 120

tan 苷

• is the reference angle. • is the angle made by the vector and the positive x-axis.

The magnitude of v is 兹65, and its direction is 120° as measured from the positive x-axis. The angle between a vector and the positive x-axis is called the direction angle of the vector. Because AB in Figure 4.26 is equivalent to v, 储 AB 储苷兹65 and the direction angle of AB is also 120°. Expressing vectors in terms of components provides a convenient method for performing operations on vectors. Definitions of Fundamental Vector Operations If v 苷具a, b典 and w 苷具c, d典 are two vectors and k is a real number, then 储 v 储苷兹a2 b2

1.

2. v w 苷具a, b典具c, d典苷具a c, b d典 3. kv 苷 k具a, b典苷具ka, kb典 In terms of components, the zero vector 0 苷具0, 0典. The additive inverse of a vector v 苷具a, b典 is given by v 苷具a, b典 . EXAMPLE 2

»

Perform Operations on Vectors

Given v 苷具2, 3典 and w 苷具4, 1典, find 储w储

a.

b.

vw

c.

3v

d.

2v 3w

Solution a.

储w储苷兹42 共1兲2 苷兹17

c.

3v 苷 3具2, 3典苷具6, 9典

b.

v w 苷具2, 3典具4, 1典苷具2 4, 3 共1兲典苷具2, 2典

d.

2v 3w 苷 2具2, 3典 3具4, 1典苷具4, 6典具12, 3典苷具16, 9典

»

Try Exercise 24, page 325

Unit Vectors A unit vector is a vector whose magnitude is 1. For example, the vector 3 4 v苷 , is a unit vector because 5 5 3 2 4 2 9 16 25 储v储苷苷苷苷1 5 5 25 25 25

冓

冔

冑冉冊冉冊冑

冑

316

Chapter 4

Applications of Trigonometry Given any nonzero vector v, we can obtain a unit vector in the direction of v by dividing each component of v by the magnitude of v, 储 v 储 . EXAMPLE 3

»

Find a Unit Vector

Find a unit vector u in the direction of v 苷具4, 2典.

Solution Find the magnitude of v. 储v储苷兹共4兲2 22 苷兹16 4 苷兹20 苷 2兹5 Divide each component of v by 储v储. u苷

冓

冔冓

4

冔冓

冔

2 2 1 2兹5 兹5 , 苷 , 苷 , . 2兹5 2兹5 5 5 兹5 兹5

A unit vector in the direction of v is u.

»

Try Exercise 12, page 325

y

2

Two unit vectors, one parallel to the x-axis and one parallel to the y-axis, are of special importance. See Figure 4.27.

1 j −2

−1

i

1

x

2

Definitions of Unit Vectors i and j

−1

i 苷具1, 0典

j 苷具0, 1典

Figure 4.27

The vector v 苷具3, 4典 can be written in terms of the unit vectors i and j as shown in Figure 4.28.

y

4 4j

v = 3i + 4j

α −4

−1

3i

Figure 4.28

4

x

具3, 4典苷具3, 0典具0, 4典苷 3具1, 0典 4具0, 1典苷 3i 4j

• Definition of vector addition • Definition of scalar multiplication of a vector • Definition of i and j

By means of scalar multiplication and addition of vectors, any vector can be expressed in terms of the unit vectors i and j. Let v 苷具a1 , a2 典. Then v 苷具a1 , a2 典苷 a1具1, 0典 a2具0, 1典苷 a1 i a2 j This gives the following result. Representation of a Vector in Terms of i and j If v is a vector and v 苷具a1 , a2 典, then v 苷 a1 i a2 j.

The definitions for addition and scalar multiplication of vectors can be restated in terms of i and j. If v 苷 a1 i a2 j and w 苷 b1 i b2 j, then v w 苷共a1 i a2 j兲共b1 i b2 j兲苷共a1 b1兲i 共a2 b2兲j kv 苷 k共a1 i a2 j兲苷 ka1 i ka2 j

4.3 EXAMPLE 4

»

Vectors

317

Operate on Vectors Written in Terms of i and j

Given v 苷 3i 4j and w 苷 5i 3j, find 3v 2w.

Solution 3v 2w 苷 3共3i 4j兲 2共5i 3j兲苷共9i 12j兲共10i 6j兲苷共9 10兲i 共12 6兲j 苷 i 18j

y

(a1, a2 ) v = 〈 a1, a2 〉

»

Try Exercise 30, page 325

θ x

Figure 4.29

The components a1 and a2 of the vector v 苷具a1 , a2 典 can be expressed in terms of the magnitude of v and the direction angle of v (the angle that v makes with the positive x-axis). Consider the vector v in Figure 4.29. Then 储 v 储苷兹共a1兲2 共a2兲2 From the definitions of sine and cosine, we have cos 苷

a1 储v储

and

sin 苷

a2 储v储

Rewriting the last two equations, we find that the components of v are a1 苷储 v 储 cos

and a2 苷储 v 储 sin

Definitions of Horizontal and Vertical Components of a Vector Let v 苷具a1 , a2 典, where v 苷 0, the zero vector. Then a1 苷储 v 储 cos

and a2 苷储 v 储 sin

where is the angle between the positive x-axis and v. The horizontal component of v is 储 v 储 cos . The vertical component of v is 储 v 储 sin .

QUESTION

Is u 苷 cos i sin j a unit vector?

Any nonzero vector can be written in terms of its horizontal and vertical components. Let v 苷 a1 i a2 j. Then v 苷 a1 i a2 j 苷共储 v 储 cos 兲i 共储 v 储 sin 兲j 苷储 v 储共cos i sin j兲 where 储 v 储 is the magnitude of v and the vector cos i sin j is a unit vector. The last equation shows that any vector v can be written as the product of its magnitude and a unit vector in the direction of v. ANSWER

Yes, because 兩兩cos i sin j兩兩苷兹cos2 sin2 苷兹1 苷 1.

318

Chapter 4

Applications of Trigonometry EXAMPLE 5

»

Find the Horizontal and Vertical Components of a Vector

Find, to the nearest tenth, the horizontal and vertical components of a vector v of magnitude 10 meters with direction angle 228°. Write the vector in the form a1i a2j.

Solution a1 苷 10 cos 228 ⬇ 6.7 a2 苷 10 sin 228 ⬇ 7.4 The approximate horizontal and vertical components are 6.7 and 7.4, respectively. v ⬇ 6.7i 7.4j

»

Try Exercise 40, page 325

Applications of Vectors take note Ground speed is the magnitude of the resultant of the plane’s velocity vector and the wind velocity vector.

Vectors are used to solve applied problems in which forces are acting simultaneously on an object. For instance, an airplane flying in a wind is being acted upon by the force of its engine, F1, and the force of the wind, F2. The combined effect of these two forces is given by the resultant F1 F2. In the following example, the airspeed of the plane is the speed at which the plane would be moving if there were no wind. The actual velocity of the plane is the plane’s velocity with respect to the ground. The magnitude of the plane’s actual velocity is called its ground speed. EXAMPLE 6

y

B 320 mph

C

α 62°

42 mph

x

D

Figure 4.30

Solve an Application Involving Airspeed

An airplane is traveling with an airspeed of 320 mph and a heading of 62°. A wind of 42 mph is blowing at a heading of 125°. Find the ground speed and the course of the airplane.

Solution

θ 125°

A

»

Sketch a diagram similar to Figure 4.30 showing the relevant vectors. AB represents the heading and the airspeed, AD represents the wind velocity, and AC represents the course and the ground speed. By vector addition, AC 苷 AB AD. From the figure, AB 苷 320共cos 28 i sin 28 j兲 AD 苷 42关cos 共35 兲i sin 共35 兲j兴 AC 苷 320共cos 28 i sin 28 j兲 42关cos 共35 兲i sin 共35 兲j兴 ⬇ 共282.5i 150.2j兲共34.4i 24.1j兲苷 316.9i 126.1j AC is the course of the plane. The ground speed is 储AC储. The heading is 苷 90 . 储AC储苷兹共316.9兲2 共126.1兲2 ⬇ 340 126.1 苷 90 苷 90 tan1 ⬇ 68 316.9

冉冊

The ground speed is approximately 340 mph at a heading of 68°.

»

Try Exercise 44, page 325

4.3

319

Vectors

We can add two vectors to produce a resultant vector, and we can also find two vectors whose sum is a given vector. The process of finding two vectors whose sum is a given vector is called resolving the vector. Many applied problems can be analyzed by resolving a vector into two vectors that are perpendicular to each other. For instance, Figure 4.31 shows a car on a ramp. The force of gravity on the car is shown by the downward vector w, which has been resolved into the vectors F1 and F2. The vector F1 is parallel to the ramp and represents the force pushing the car down the ramp. The vector F2 is perpendicular to the ramp and represents the force the car exerts against the ramp. These two perpendicular forces F1 and F2 are vector components of w. That is, w F1 F2. The force needed to keep the car from rolling down the ramp is F1. The angle formed by w and F2 is complementary to the 75.0º angle. Thus the angle formed by w and F2 has a measure of 15.0º, which is the same as the angular measure of the incline of the ramp.

75.0°

15.0°

15.0° F2

w −F1 F1

Figure 4.31

EXAMPLE 7

»

Solve an Application Involving Force

The car shown in Figure 4.31 has a weight of 2855 pounds. a. b.

Find the magnitude of the force needed to keep the car from rolling down the ramp. Round to the nearest pound. Find the magnitude of the force the car exerts against the ramp. Round to the nearest pound.

Solution a.

The force needed to keep the car from rolling down the ramp is represented by –F1 in Figure 4.31. The magnitude of –F1 equals the magnitude of F1. The triangle formed by w, F1, and F2 is a right triangle in 储 F1 储 which sin 15.0º Use this equation to find 储 F 1 储 . 储w储储F1储储w储储F1储 sin 15.0 苷 • 储 w 储 2855 2855 • Solve for 储 F1 储 . 储F1储苷 2855 sin 15.0 ⬇ 739 Approximately 739 pounds of force is needed to keep the car from rolling down the ramp. sin 15.0 苷

Continued

䉴

320

Chapter 4

Applications of Trigonometry b. The vector F2 in Figure 4.31 is the force the car exerts against the ramp. The triangle formed by w, F1, and F2 is a right triangle in which 储 F2 储 cos 15.0º . Use this equation to find 储 F2 储 . 储w储储F2储储w储储F2储 cos 15.0 苷 2855 储F2储苷 2855 cos 15.0 ⬇ 2758

cos 15.0 苷

• 储 w 储 2855 • Solve for 储 F2 储 .

The magnitude of the force the car exerts against the ramp is approximately 2758 pounds.

»

Try Exercise 48, Page 326

Dot Product We have considered the product of a real number (scalar) and a vector. We now turn our attention to the product of two vectors. Finding the dot product of two vectors is one way to multiply a vector by a vector. The dot product of two vectors is a real number and not a vector. The dot product is also called the inner product or the scalar product. This product is useful in engineering and physics. Definition of Dot Product Given v 苷具a, b典 and w 苷具c, d典, the dot product of v and w is given by v w 苷 ac bd

EXAMPLE 8

»

Find the Dot Product of Two Vectors

Find the dot product of v 苷具6, 2典 and w 苷具3, 4典.

Solution v w 苷 6共3兲共2兲4 苷 18 8 苷 26

»

Try Exercise 60, page 326

QUESTION

Is the dot product of two vectors a vector or a real number?

If the vectors in Example 8 were given in terms of the vectors i and j, then v 苷 6i 2j and w 苷 3i 4j. In this case, v w 苷共6i 2j兲共3i 4j兲苷 6共3兲共2兲4 苷 26

ANSWER

A real number

4.3

Vectors

321

Properties of the Dot Product In the following properties, u, v, and w are vectors and a is a scalar. 1. v w 苷 w v

2. u 共v w兲苷 u v u w

3. a共u v兲苷共au兲 v 苷 u 共av兲

4. v v 苷储 v 储 2

5. 0 v 苷 0

6.

ii苷jj苷1

7. i j 苷 j i 苷 0

The proofs of these properties follow from the definition of dot product. Here is the proof of the fourth property. Let v 苷 ai bj. v v 苷共ai bj兲共ai bj兲苷 a2 b 2 苷储 v 储 2 Rewriting the fourth property of the dot product yields an alternative way of expressing the magnitude of a vector.

Magnitude of a Vector in Terms of the Dot Product If v 苷具a, b典, then 储 v 储苷兹v v.

y

The Law of Cosines can be used to derive an alternative formula for the dot product. Consider the vectors v 苷具a, b典 and w 苷具c, d典 as shown in Figure 4.32. Using the Law of Cosines for triangle OAB, we have

A(a, b) AB B(c, d)

v

α

储 AB 储 2 苷储 v 储 2 储 w 储 2 2 储 v 储储 w 储 cos

w

O(0, 0)

Figure 4.32

x

By the distance formula, 储 AB 储 2 苷共a c兲2 共b d兲2, 储 v 储 2 苷 a2 b 2, and 储 w 储 2 苷 c2 d 2. Thus 共a c兲2 共b d兲2 苷共a2 b 2 兲共c2 d 2 兲 2 储 v 储储 w 储 cos a a2 2ac c 2 b 2 2bd d 2 苷 a2 b 2 c2 d 2 2 储 v 储储 w 储 cos a 2ac 2bd 苷 2 储 v 储储 w 储 cos a ac bd 苷储 v 储储 w 储 cos a v w 苷储 v 储储 w 储 cos a

• v • w ac bd

Alternative Formula for the Dot Product If v and w are two nonzero vectors and is the smallest nonnegative angle between v and w, then v w 苷储 v 储储 w 储 cos .

Solving the alternative formula for the dot product for cos , we have a formula for the cosine of the angle between two vectors.

322

Chapter 4

Applications of Trigonometry Angle Between Two Vectors If v and w are two nonzero vectors and a is the smallest nonnegative vw vw angle between v and w, then cos 苷 and 苷 cos1 . 储v储储w储储v储储w储

冉

EXAMPLE 9 y

»

冊

Find the Angle Between Two Vectors

Find, to the nearest tenth of a degree, the measure of the smallest nonnegative angle between the vectors v 苷 2i 3j and w 苷 i 5j, as shown in

4

Figure 4.33.

w

α

Solution 4 x

−4 v

Use the equation for the angle between two vectors. vw 共2i 3j兲共i 5j兲苷储v储储w储共兹22 共3兲2兲共兹共1兲2 52兲 2 15 17 苷苷兹13 兹26 兹338 17 苷 cos1 ⬇ 157.6 兹338

cos 苷

−4

Figure 4.33

冉冊

The measure of the smallest nonnegative angle between the two vectors is approximately 157.6°.

»

Try Exercise 70, page 326

v

α

Scalar Projection

w

proj wv

α

v

w

Let v 苷具a1 , a2 典 and w 苷具b1 , b2 典 be two nonzero vectors, and let be the angle between the vectors. Two possible configurations, one for which is an acute angle and one for which is an obtuse angle, are shown in Figure 4.34. In each case, a right triangle is formed by drawing a line segment from the terminal point of v to a line through w. Definition of the Scalar Projection of v onto w

projwv

Figure 4.34

If v and w are two nonzero vectors and is the angle between v and w, then the scalar projection of v onto w, projw v, is given by projwv 苷储 v 储 cos

To derive an alternate formula for projw v , consider the dot product v w 苷储 v 储储 w 储 cos . Solving for 储 v 储 cos , which is projw v, we have vw projw v 苷储w储 When the angle between the two vectors is an acute angle, projw v is positive. When is an obtuse angle, projw v is negative.

4.3 y

EXAMPLE 10

»

Vectors

323

Find the Projection of v onto w

Given v 苷 2i 4j and w 苷 2i 8j as shown in Figure 4.35, find projwv.

5 w

Solution α

proj wv

v

Use the equation projwv 苷 5 x

projw v 苷

Figure 4.35

vw . 储w储

共2i 4j兲共2i 8j兲 28 14兹17 苷苷 ⬇ 3.4 2 2 17 兹共2兲 8 兹68

»

Try Exercise 72, page 326

Parallel and Perpendicular Vectors α = 0°

α = 180°

Figure 4.36

90°

Figure 4.37

Two vectors are parallel when the angle between the vectors is 0° or 180°, as shown in Figure 4.36. When the angle is 0°, the vectors point in the same direction; the vectors point in opposite directions when is 180°. Let v 苷 a1 i b1 j, let c be a real number, and let w 苷 cv. Because w is a constant multiple of v, w and v are parallel vectors. When c 0 , the vectors point in the same direction. When c 0, the vectors point in opposite directions. Two vectors are perpendicular when the angle between the vectors is 90°. See Figure 4.37. Perpendicular vectors are referred to as orthogonal vectors. If v and w are two nonzero orthogonal vectors, then from the formula for the angle between two vectors and the fact that cos 苷 0, we have 0苷

vw 储v储储w储

If a fraction equals zero, the numerator must be zero. Thus, for orthogonal vectors v and w, v w 苷 0. This gives the following result.

Condition for Perpendicular Vectors Two nonzero vectors v and w are orthogonal if and only if v w 苷 0.

Work = 20 ft-lb 4 ft

5 lb

Figure 4.38

Work: An Application of the Dot Product When a 5-pound force is used to lift a box from the ground a distance of 4 feet, work is done. The amount of work is the product of the force on the box and the distance the box is moved. In this case the work is 20 foot-pounds. When the box is lifted, the force and the displacement vector (the direction in which and the distance the box was moved) are in the same direction. (See Figure 4.38.)

324

Chapter 4

Applications of Trigonometry Now consider a sled being pulled by a child along the ground by a rope attached to the sled, as shown in Figure 4.39. The force vector (along the rope) is not in the same direction as the displacement vector (parallel to the ground). In this case the dot product is used to determine the work done by the force. Definition of Work The work W done by a force F applied along a displacement s is W 苷 F s 苷兩兩F兩兩兩兩s兩兩 cos a

25 lb 37°

Figure 4.39

where is the angle between F and s.

In the case of the child pulling the sled a horizontal distance of 7 feet, the work done is W苷Fs 苷储 F 储储 s 储 cos • is the angle between F and s. 苷共25兲共7兲 cos 37 ⬇ 140 foot-pounds

50 lb 37°

10°

Figure 4.40

EXAMPLE 11

»

Solve a Work Problem

A force of 50 pounds on a rope is used to drag a box up a ramp that is inclined 10°. If the rope makes an angle of 37° with the ground, find the work done in moving the box 15 feet along the ramp. See Figure 4.40.

Solution In the formula W 苷储F储储s储 cos a, a is the angle between the force and the displacement. Thus 苷 37 10 苷 27 . The work done is W 苷储F储储s储 cos a 苷 50 15 cos 27 ⬇ 670 foot-pounds

»

Try Exercise 80, page 326

Topics for Discussion 1. Is the dot product of two vectors a vector or a scalar? Explain. 2. Is the projection of v onto w a vector or a scalar? Explain. 3. Is the nonzero vector 具a, b典 perpendicular to the vector 具b, a典? Explain. 4. Consider the nonzero vector u 苷具a, b典 and the vector v苷 a.

冓

冔

a b , 2 2 兹a b 兹a b2 2

Are the vectors parallel? Explain.

b. Which one of the vectors is a unit vector? c.

Which vector has the larger magnitude? Explain.

4.3

325

Vectors

Exercise Set 4.3 In Exercises 1 to 10, find the components of the vector with the initial point P1 and terminal point P2. Use these components to write a vector that is equivalent to P1 P2.

1.

y

2.

P1

P2 2

2

P1

−2 2

3.

y

P2

4

x

P1

32.

34. 储 v 储

35. 储 u 2v 储

1 3 u v 2 4

33.

3 2 v u 3 4

36. 储 2v 3u 储

37. Magnitude 苷 5, direction angle 苷 27

4 P2

2

38. Magnitude 苷 4, direction angle 苷 127

2 P1

−2

31. 6u 2v

30. 3u 2v

In Exercises 37 to 40, find the horizontal and vertical components of each vector. Round to the nearest tenth. Write an equivalent vector in the form v a1 i a2 j.

y

4.

4

29. 4v

P2

x

4

28. 2u

y 2

4

In Exercises 28 to 36, perform the indicated operations, where u 3i 2j and v 2i 3j.

2

x

−2

2

5. P1共3, 0兲; P2共4, 1兲

6. P 1共5, 1兲; P 2共3, 1兲

7. P1共4, 2兲; P2共3, 3兲

8. P1共0, 3兲; P2共0, 4兲

9. P1共2, 5兲; P2共2, 3兲

In Exercises 11 to 18, find the magnitude and direction of each vector. Find the unit vector in the direction of the given vector.

12. v 苷具6, 10典

13. v 苷具20, 40典

14. v 苷具50, 30典

15. v 苷 2i 4j

16. v 苷 5i 6j

17. v 苷 42i 18j

18. v 苷 22i 32j

In Exercises 19 to 27, perform the indicated operations, where u 具2, 4典 and v 具3, 2典.

19. 3u

20. 4v

22. 4v 2u

23.

25. 储 u 储

26. 储 v 2u 储

2 1 u v 3 6

21. 2u v 24.

4

40. Magnitude 苷 2, direction angle 苷

8 7

4 x

10. P1共3, 2兲; P2共3, 0兲

11. v 苷具3, 4典

39. Magnitude 苷 4, direction angle 苷

3 u 2v 4

27. 储 3u 4v 储

41. GROUND SPEED OF A PLANE A plane is flying at an airspeed of 340 mph at a heading of 124°. A wind of 45 mph is blowing from the west. Find the ground speed of the plane. 42. HEADING OF A BOAT A person who can row 2.6 mph in still water wants to row due east across a river. The river is flowing from the north at a rate of 0.8 mph. Determine the heading of the boat required for the boat to travel due east across the river. 43. GROUND SPEED AND COURSE OF A PLANE A pilot is flying at a heading of 96° at 225 mph. A 50-mph wind is blowing from the southwest at a heading of 37°. Find the ground speed and course of the plane. 44. COURSE OF A BOAT The captain of a boat is steering at a heading of 327° at 18 mph. The current is flowing at 4 mph at a heading of 60°. Find the course (to the nearest degree) of the boat. 45. MAGNITUDE OF A FORCE Find the magnitude of the force necessary to keep a 3000-pound car from sliding down a ramp inclined at an angle of 5.6°. Round to the nearest pound.

326

Chapter 4

Applications of Trigonometry

46. ANGLE OF A RAMP A 120-pound force keeps an 800-pound object from sliding down an inclined ramp. Find the angle of the ramp. Round to the nearest tenth of a degree. 47. MAGNITUDE OF A FORCE A 345-pound box is placed on a ramp that is inclined 22.4º. a. Find the magnitude of the force needed to keep the box from sliding down the ramp. Ignore the effects of friction. Round to the nearest pound. b. Find the magnitude of the force the box exerts against the ramp. Round to the nearest pound. 48. MAGNITUDE OF A FORCE A motorcycle that weighs 811 pounds is placed on a ramp that is inclined 31.8º. a. Find the magnitude of the force needed to keep the motorcycle from rolling down the ramp. Round to the nearest pound. b. Find the magnitude of the force the motorcycle exerts against the ramp. Round to the nearest pound.

In Exercises 55 to 62, find the dot product of the vectors.

55. v 苷具3, 2典; w 苷具1, 3典

56. v 苷具2, 4典; w 苷具0, 2典

57. v 苷具4, 1典; w 苷具1, 4典

58. v 苷具2, 3典; w 苷具3, 2典

59. v 苷 i 2j; w 苷 i j

60. v 苷 5i 3j; w 苷 4i 2j

61. v 苷 6i 4j; w 苷 2i 3j 62. v 苷 4i 2j; w 苷 2i 4j In Exercises 63 to 70, find the measure of the angle between the two vectors. State which pairs of vectors are orthogonal. Round approximate measures to the nearest tenth of a degree.

63. v 苷具2, 1典; w 苷具3, 4典

64. v 苷具1, 5典; w 苷具2, 3典

65. v 苷具0, 3典; w 苷具2, 2典

66. v 苷具1, 7典; w 苷具3, 2典

67. v 苷 5i 2j; w 苷 2i 5j

68. v 苷 8i j; w 苷 i 8j

69. v 苷 5i 2j; w 苷 5i 2j The forces F1, F2, F3, . . . , Fn acting on an object are in equilibrium provided the resultant of all the forces is the zero vector: F1 F2 F3 Fn 0 In Exercises 49 to 53, determine whether the given forces are in equilibrium. If the forces are not in equilibrium, determine an additional force that would bring the forces into equilibrium.

49. F1 苷具18.2, 13.1典, F2 苷具 12.4, 3.8典, F3 苷具 5.8, 16.9典 50. F1 苷具 4.6, 5.3典, F2 苷具6.2, 4.9典, F3 苷具1.6, 10.2典 51. F1 苷 155i 257j, F2 苷 124i 149j, F3 苷 31i 98j 52. F1 苷 23.5i 18.9j, F2 苷 18.7i 2.5j, F3 苷 5.6i 15.6j

70. v 苷 3i 4j; w 苷 6i 12j In Exercises 71 to 78, find projw v.

71. v 苷具6, 7典; w 苷具3, 4典

72. v 苷具7, 5典; w 苷具4, 1典

73. v 苷具3, 4典; w 苷具2, 5典

74. v 苷具2, 4典; w 苷具1, 5典

75. v 苷 2i j; w 苷 6i 3j 76. v 苷 5i 2j; w 苷 5i 2j 77. v 苷 3i 4j; w 苷 6i 12j 78. v 苷 2i 2j; w 苷 4i 2j

53. F1 苷 189.3i 235.7j, F2 苷 45.8i 205.6j, F3 苷 175.2i 37.7j, F4 苷 59.9i 7.6j

79. WORK A 150-pound box is dragged 15 feet along a level floor. Find the work done if a force of 75 pounds with a direction angle of 32° is used. Round to the nearest foot-pound.

54. The cranes in the following figure are holding a 9450pound steel girder in midair. The forces F1 and F2, along with the force F3 苷具0, 9450典, are in equilibrium as defined above Exercise 49. Find the magnitude of F2, given that ||F1|| 6223 pounds. Round to the nearest 10 pounds.

80. WORK A 100-pound force is pulling a sled loaded with bricks that weighs 400 pounds. The force is at an angle of 42° with the displacement. Find the work done in moving the sled 25 feet.

y

F2

F1

36°

81. WORK A rope is being used to pull a box up a ramp that is inclined at 15°. The rope exerts a force of 75 pounds on the box, and it makes an angle of 30° with the plane of the ramp. Find the work done in moving the box 12 feet.

49° x

82. WORK A dock worker exerts a force on a box sliding down the ramp of a truck. The ramp makes an angle of 48° with the road, and the worker exerts a 50-pound force parallel to the road. Find the work done in sliding the box 6 feet.

4.3

327

Vectors

Connecting Concepts 83. For u 苷具1, 1典, v 苷具2, 3典, and w 苷具5, 5典, find the sum of the three vectors geometrically by using the triangle method of adding vectors. 84. For u 苷具1, 2典, v 苷具3, 2典, and w 苷具1, 4典, find u v w geometrically by using the triangle method of adding vectors. 85. Find a vector that has initial point 共3, 1兲 and is equivalent to v 苷 2i 3j. 86. Find a vector that has initial point 共2, 4兲 and is equivalent to v 苷具1, 3典. If v 苷 2i 5j and w 苷 5i 2j have the same initial point, is v perpendicular to w? Why or why

87.

93. Prove that c共v w兲苷共cv兲 w. 94. Show that the dot product of two nonzero vectors is positive if the angle between the vectors is an acute angle and negative if the angle between the two vectors is an obtuse angle. 95. COMPARISON OF WORK DONE Consider the following two situations. (1) A rope is being used to pull a box up a ramp inclined at an angle a. The rope exerts a force F on the box, and the rope makes an angle u with the ramp. The box is pulled s feet. (2) A rope is being used to pull the same box along a level floor. The rope exerts the same force F on the box. The box is pulled the same s feet. In which case is more work done?

not? 88.

If v 苷具5, 6典 and w 苷具6, 5典 have the same initial point, is v perpendicular to w? Why or why not?

s F

89. Let v 苷具2, 7典. Find a vector perpendicular to v. 90. Let w 苷 4i j. Find a vector perpendicular to w.

(1)

θ

α

91. Is the dot product an associative operation? That is, given any nonzero vectors u, v, and w, does

F

共u v兲 w 苷 u 共v w兲?

θ

(2)

s

92. Prove that v w 苷 w v.

Projects 1. SAME DIRECTION OR OPPOSITE DIRECTIONS Let v 苷 cw, where c is a nonzero real number and w is a nonzero vecvw 苷 1 and that the result is 1 tor. Show that 储v储储w储 when c 0 and 1 when c 0. 2. THE LAW OF COSINES AND VECTORS Prove that 储 v w 储 2 苷储 v 储 2 储 w 储 2 2v w

3. PROJECTION RELATIONSHIPS Determine the relationship between the nonzero vectors v and w if a. projw v 苷 0

b. projw v 苷储 v 储

328

Chapter 4

Applications of Trigonometry

Exploring Concepts with Technology Optimal Branching of Arteries The physiologist Jean Louis Poiseuille (1799–1869) developed several laws concerning the flow of blood. One of his laws states that the resistance R of a blood vessel of length l and radius r is given by R苷k

r2

P1

r1

P3

冉

R苷k

θ

a

Figure 4.41

(1)

The number k is a variation constant that depends on the viscosity of the blood. Figure 4.41 shows a large artery with radius r1 and a smaller artery with radius r2 . The branching angle between the arteries is . Make use of Poiseuille’s Law, Equation (1), to show that the resistance R of the blood along the path P1 P2 P3 is b

P2

l r4

冊

a b cot u b csc u (r1)4 (r2)4

(2)

Use a graphing utility to graph R with k 苷 0.0563, a 苷 8 centimeters, b 苷 4 centi3 meters, r1 苷 0.4 centimeter, and r2 苷 r1 苷 0.3 centimeter. Then estimate (to the 4 nearest degree) the angle that minimizes R. By using calculus, it can be demonstrated that R is minimized when cos 苷

冉冊 r2 r1

4

(3)

This equation is remarkable because it is much simpler than Equation (2) and because it does not involve the distance a or b. Solve Equation (3) for , with 3 r2 苷 r1 . How does this value of compare with the value of you obtained 4 by graphing?

Chapter 4 Summary 4.1 The Law of Sines • The following Law of Sines is used to solve triangles when two angles and a side are given or when two sides and an angle opposite one of them are given. a b c 苷苷 sin A sin B sin C

• The area K of triangle ABC is K苷

1 b2 sin C sin A bc sin A 苷 2 2 sin B

• The area of a triangle for which three sides are given (Heron’s formula) is K 苷兹s共s a兲共s b兲共s c兲,

4.2 The Law of Cosines and Area • The Law of Cosines, a2 苷 b 2 c2 2bc cos A, is used to solve general triangles when two sides and the included angle or three sides of the triangle are given.

where s 苷

1 共a b c兲 2

4.3 Vectors • A vector is a directed line segment. The length of the line segment is the magnitude of the vector, and the direction of

Review Exercises the vector is measured by an angle. The angle between a vector and the positive x-axis is called the direction angle of the vector. Two vectors are equivalent if they have the same magnitude and the same direction. The resultant of two or more vectors is the sum of the vectors. • Vectors can be added by the parallelogram method, the triangle method, or addition of the x- and y-components.

329

• If v 苷具a, b典 and k is a real number, then kv 苷具ka, kb典. • The dot product of v 苷具a, b典 and w 苷具c, d典 is given by v w 苷 ac bd • If v and w are two nonzero vectors and a is the smallest vw . nonnegative angle between v and w, then cos a 苷储v储储w储

Chapter 4 Assessing Concepts 1. What is an oblique triangle? 2. In triangle ABC, a 4.5, b 6.2, and C 107º. Which law, the Law of Sines or the Law of Cosines, can be used to find c? 3. Which of the following cases, ASA, AAS, SSA, SSS, or SAS is known as the ambiguous case of the Law of Sines? 4. In Heron’s formula, what does the variable s represent?

6. Let v and w be nonzero vectors. Is projwv a vector or a scalar? 7. True or false: The vector

冓

冔

12 5 is a unit vector. , 13 13

8. True or false: i j 0. 9. True or false: The Law of Sines can be used to solve any triangle, given two angles and any side. 10. True or false: If two nonzero vectors are orthogonal, then their dot product is 0.

5. Is the dot product of two vectors a vector or a scalar?

Chapter 4 Review Exercises In Exercises 1 to 10, solve each triangle.

1. A 苷 37 , b 苷 14, C 苷 92 2. B 苷 77.4 , c 苷 11.8, C 苷 94.0 3. a 苷 12, b 苷 15, c 苷 20 4. a 苷 24, b 苷 32, c 苷 28 5. a 苷 18, b 苷 22, C 苷 35 6. b 苷 102, c 苷 150, A 苷 82 7. A 苷 105 , a 苷 8, c 苷 10 8. C 苷 55 , c 苷 80, b 苷 110 9. A 苷 55 , B 苷 80 , c 苷 25 10. B 苷 25 , C 苷 40 , c 苷 40

In Exercises 11 to 18, find the area of each triangle. Round each area accurate to two significant digits.

11. a 苷 24, b 苷 30, c 苷 36

12. a 苷 9.0, b 苷 7.0, c 苷 12

13. a 苷 60, b 苷 44, C 苷 44

14. b 苷 8.0, c 苷 12, A 苷 75

15. b 苷 50, c 苷 75, C 苷 15

16. b 苷 18, a 苷 25, A 苷 68

17. A 苷 110 , a 苷 32, b 苷 15

18. A 苷 45 , c 苷 22, b 苷 18

In Exercises 19 and 20, find the components of each vector with the given initial and terminal points. Write an equivalent vector in terms of its components.

19. P1共2, 4兲; P2共3, 7兲

20. P1共4, 0兲; P2共3, 6兲

In Exercises 21 to 24, find the magnitude and direction angle of each vector.

21. v 苷具4, 2典

22. v 苷具6, 3典

23. u 苷 2i 3j

24. u 苷 4i 7j

330

Chapter 4

Applications of Trigonometry

In Exercises 25 to 28, find a unit vector in the direction of the given vector.

25. w 苷具8, 5典

26. w 苷具7, 12典

27. v 苷 5i j

28. v 苷 3i 5j

In Exercises 35 to 38, find the dot product of the vectors.

35. u 苷具3, 7典; v 苷具1, 3典 36. v 苷具8, 5典; u 苷具2, 1典 37. v 苷 4i j; u 苷 2i j

In Exercises 29 and 30, perform the indicated operation, where u 具3, 2典 and v 具4, 1典.

38. u 苷 3i 7j; v 苷 2i 2j

29. v u

In Exercises 39 to 42, find the angle between the vectors. Round to the nearest degree.

30. 2u 3v

In Exercises 31 and 32, perform the indicated operation, where u 10i 6j and v 8i 5j.

2 3 32. v u 3 4

1 31. u v 2

39. u 苷具7, 4典; v 苷具2, 3典 40. v 苷具5, 2典; u 苷具2, 4典 41. v 苷 6i 11j; u 苷 2i 4j

33. GROUND SPEED OF A PLANE A plane is flying at an airspeed of 400 mph at a heading of 204°. A wind of 45 mph is blowing from the east. Find the ground speed of the plane. 34. ANGLE OF A RAMP A 40-pound force keeps a 320-pound object from sliding down an inclined ramp. Find the angle of the ramp.

42. u 苷 i 5j; v 苷 i 5j In Exercises 43 and 44, find projw v.

43. v 苷具2, 5典; w 苷具5, 4典 44. v 苷 4i 7j; w 苷 2i 5j 45. WORK A 120-pound box is dragged 14 feet along a level floor. Find the work done if a force of 60 pounds with a direction angle of 38° is used. Round to the nearest foot-pound.

LATITU DE

» » » Quantitative Reasoning: Trigonometry and Great Circle Routes »»»

LONGITU

DE

E Q U A T O R

Great circle route from New York’s John F. Kennedy airport to London’s Heathrow airport.

A great circle is a circle on a sphere’s surface whose center is the same as the center of the sphere. A great circle is a path on the sphere with the smallest curvature, and hence an arc of a great circle is the shortest path between two given points on the sphere. The distance between any two points on a sphere is called the great circle distance for the given points. Great circle routes are often used by pilots when winds are not a significant factor. A formula that is used to estimate the great circle distance d, measured in radians, between two airports is given by d 苷 cos1[cos(lat1) cos(lat2) cos(lon1lon2) sin(lat1) sin(lat2)]

(1)

where lat1 is the latitude of airport 1, lon1 is the longitude of airport 1, lat2 is the latitude of airport 2, and lon2 is the longitude of airport 2. For example, the coordinates of New York’s John F. Kennedy airport (JFK) are 40º39 N and 73º47 W. The coordinates of London’s Heathrow airport (LHR) are 51º29 N and 0º27 W. Converting each of these coordinates to radians gives (lat1, lon1) ⬇ (0.709476, 1.287756) (lat2, lon2) ⬇ (0.898554, 0.007854)

Quantitative Reasoning

331

Note: North latitudes and east longitudes are represented by positive radian values, whereas south latitudes and west longitudes are represented by negative radian values. Substitute the radian values into Formula (1) to produce d ⬇ 0.869496 radian. Because the earth is nearly spherical with a radius of about 3960 miles, we can convert d to miles by multiplying by 3960. d ⬇ 0.869496 3960 ⬇ 3443 Rounded to the nearest 10 miles, the great circle distance from JFK to LHR is approximately 3440 miles. To fly a great circle route, the pilot must adjusts his or her heading several times during the flight. The following formulas can be used to find the initial heading (h1) a pilot should use to fly a great circle route from airport 1 to airport 2. If sin(lon2 – lon1) > 0, then h1 苷 cos1

冋

sin(lat2) sin(lat1) cos(d) sin(d) cos(lat1)

Otherwise, h1 苷 2p cos1

冋

册

sin(lat2) sin(lat1) cos(d) sin(d) cos(lat1)

(2)

册

(3)

In Formulas (2) and (3), each of the values lat1, lat2, and d must be in radians. Also, these formulas are not valid if airport 1 is located at the North or South Pole. To find the initial heading required to fly the great circle route from JFK to LHR, we first note that sin(lon2 lon1) 苷 sin[0.007854 (1.287756)] ⬇ 0.957988 Because sin(lon2 lon1) 0, we use Formula (2) to find the initial heading. Substitute 0.709476 for lat1, 0.898554 for lat2, and 0.869496 for d to produce h1 ⬇ 0.896040 radian. Converting to degrees gives 51º as the initial heading from JFK to LHR, to the nearest degree. QR1. Find the great circle distance, in miles, between Orlando International airport (KMCO) (latitude 0.496187 radian, longitude 1.419110 radians) and Los Angeles International airport (LAX) (latitude 0.592409 radian, longitude 2.066611 radians). Use 3960 miles as the radius of Earth. Round to the nearest 10 miles. QR2. Find the initial heading needed to fly the great circle route from KMCO to LAX. Round to the nearest degree. QR3. Find the great circle distance, in miles, between JFK and Denver International airport (KDEN) (latitude 0.695717 radian, longitude 1.826892 radians). Use 3960 miles as the radius of Earth. Round to the nearest 10 miles. QR4. Find the initial heading required to fly the great circle route from KDEN to JFK. Round to the nearest degree.

332

Chapter 4

Applications of Trigonometry

Chapter 4 Test 1. Solve triangle ABC if A 苷 70 , C 苷 16 , and c 苷 14. 2. Find B in triangle ABC if A 苷 140 , b 苷 13, and a 苷 45. 3. In triangle ABC, C 苷 42 , a 苷 20, and b 苷 12. Find side c. 4. In triangle ABC, a 苷 32, b 苷 24, and c 苷 18. Find angle B. In Exercises 5 to 7, round your answers to two significant digits.

5. Given angle C 苷 110 , side a 苷 7.0, and side b 苷 12, find the area of triangle ABC. 6. Given angle B 苷 42 , angle C 苷 75 , and side b 苷 12, find the area of triangle ABC. 7. Given side a 苷 17, side b 苷 55, and side c 苷 42, find the area of triangle ABC. 8. Given v 2i 3j, find v.

10. Find 3u 5v given the vectors u 苷 2i 3j and v 苷 5i 4j. 11. Find the dot product of u 苷 2i 3j and v 苷 5i 3j. 12. Find the smallest positive angle, to the nearest degree, between the vectors u 苷具3, 5典 and v 苷具6, 2典. 13. One ship leaves a port at 1:00 P.M. traveling at 12 mph at a heading of 65°. At 2:00 P.M. another ship leaves the port traveling at 18 mph at a heading of 142°. Find the distance between the ships at 3:00 P.M. 14. Two fire lookouts are located 12 miles apart. Lookout A is at a bearing of N32°W from lookout B. A fire was sighted at a bearing of S82°E from A and N72°E from B. Find the distance of the fire from lookout B. 15. A triangular commercial piece of real estate is priced at $8.50 per square foot. Find the cost, to the nearest $100, of the lot, which measures 112 feet by 165 feet by 140 feet.

9. A vector has a magnitude of 12 and direction 220°. Write an equivalent vector in the form v 苷 a1 i a2 j. Round a1 and a2 to four significant digits.

Cumulative Review Exercises 1. Find the distance between P1共3, 4兲 and P2共4, 1兲.

8. Graph y 苷

2. Given f 共x兲苷 cos x and g共x兲苷 sin x, find 共 f g兲共x兲.

1 tan 2x. 4

9. Graph y 苷 2 sin共px兲 1.

3. Given f 共x兲苷 sec x and g共x兲苷 cos x, find 共 f ⴰ g兲共x兲. 1 4. Given f共x兲苷 x 3, find f 1共x兲. 2

10. Find the amplitude, period, and phase shift of the graph 1 p of y 苷 3 sin x . 3 2

5. How is the graph of F共x兲苷 f共x 2兲 3 related to the graph of y 苷 f共x兲?

11. Find the amplitude, period, and phase shift of the graph of y 苷 sin x cos x.

6. For the right triangle shown at the right, find a.

12. Find c for the triangle at the right.

a

27°

7. Graph y 苷 3 sin px.

冉

15 cm

冊

c

10 cm

50° 12 cm

Cumulative Review Exercises 13. Verify the identity

1 cos x 苷 sin x tan x. cos x

冉冉冊冊冉冉冊冊

14. Evaluate sin1 sin

2p 3

.

15. Evaluate tan cos1

12 13

.

16. Solve sin x tan x

1 tan x 苷 0 for 0 x 2p. 2

17. Find the magnitude and the positive direction angle for the vector 具4, 3典. Round the angle to the nearest tenth of a degree.

333

18. Find the angle between the vectors v 苷具1, 2典 and w 苷具2, 3典. Round to the nearest tenth of a degree. 19. HEADING OF A BOAT A person who can row at 3 mph in still water wants to row due west across a river. This river is flowing north to south at a rate of 1 mph. Determine the heading of the boat that is required to travel due west across the river. 20. GROUND SPEED AND COURSE OF A PLANE An airplane is traveling with an airspeed of 515 mph at a heading of 54.0°. A wind of 150 mph is blowing at a heading of 120.0°. Find the ground speed and the course of the plane.

5

Complex Numbers 5.1 Complex Numbers 5.2 Trigonometric Form of Complex Numbers 5.3 De Moivre’s Theorem

Fractals

A fractal image produced on a computer. As you zoom in on a region, additonal details of the fractal are displayed.

Fractal images can be produced using complex numbers and simple algebraic procedures.

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT6e

334

A fractal is a geometric figure that reveals greater detail and complexity as it is magnified over and over. Traditional Euclidean figures, such as a circle, a parabola, and a hyperbola, appear simpler as they are magnified. For instance, if you use a computer to repeatedly zoom in on a region of one of these figures, the figure starts to appear more and more like a straight line. However, as you repeatedly zoom in on a region of a fractal, more and more complex details of the figure are displayed. The word fractal was first used by the mathematician Benoit B. Mandelbrot (1924– ). Earlier mathematicians such as Cantor, Lebesgue, Julia, Koch, Hausdorff, Peano, Bolzano, and Sierpinski developed many concepts regarding fractals; however, these ideas were largely neglected until Mandelbrot publicized The Fractal Geometry of Nature in 1977. Many fractals are generated using complex numbers and a feedback process. The process starts by substituting a complex number into a complex function to produce an output. The output is substituted back into the function and the process is repeated. The Exploring Concepts with Technology on page 356 and the Quantitative Reasoning feature on page 359 explain the details of how a fractal called the Mandelbrot set is generated and how you can use a graphing calculator to produce its graph.

5.1

Section 5.1 ■

■

■

■

■

Introduction to Complex Numbers Addition and Subtraction of Complex Numbers Multiplication of Complex Numbers Division of Complex Numbers Powers of i

The Real Number System

335

Complex Numbers Introduction to Complex Numbers Recall that 兹9 苷 3 because 32 苷 9. Now consider the expression 兹9. To find 兹9, we need to find a number c such that c 2 苷 9. However, the square of any real number c (except zero) is a positive number. Consequently, we must expand our concept of number to include numbers whose squares are negative numbers. Around the seventeenth century, a new number, called an imaginary number, was defined so that a negative number would have a square root. The letter i was chosen to represent the number whose square is 1. Definition of i

Math Matters It may seem strange to just invent new numbers, but that is how mathematics evolves. For instance, negative numbers were not an accepted part of mathematics until well into the thirteenth century. In fact, these numbers often were referred to as “fictitious numbers.” In the seventeenth century, Rene Descartes called square roots of negative numbers “imaginary numbers,” an unfortunate choice of words, and started using the letter i to denote these numbers. These numbers were subjected to the same skepticism as negative numbers. It is important to understand that these numbers are not imaginary in the dictionary sense of the word. This misleading word is similar to the situation of negative numbers being called fictitious. If you think of a number line, then the numbers to the right of zero are positive numbers and the numbers to the left of zero are negative numbers. One way to think of an imaginary number is to visualize it as up or down from zero. See the Project on page 342 for more information on this topic.

The imaginary unit, designated by the letter i, is the number such that i 2 苷 1.

The principal square root of a negative number is defined in terms of i. Definition of an Imaginary Number If a is a positive real number, then 兹a 苷 i兹a. The number i兹a is called an imaginary number. Example

兹36 苷 i兹36 苷 6i 兹23 苷 i兹23

兹18 苷 i兹18 苷 3i兹2 兹1 苷 i兹1 苷 i

It is customary to write i in front of a radical sign, as we did for i兹23, to avoid confusing 兹a i with 兹ai. Definition of a Complex Number A complex number is a number of the form a bi, where a and b are real numbers and i 苷兹1. The number a is the real part of a bi, and b is the imaginary part. Example

3 5i 2 6i 5 7i

• Real part: 3; imaginary part: 5 • Real part: 2; imaginary part: 6 • Real part: 5; imaginary part: 0 • Real part: 0; imaginary part: 7

Note from these examples that a real number is a complex number whose imaginary part is zero, and an imaginary number is a complex number whose real part is zero, and whose imaginary part is not zero.

336

Chapter 5

Complex Numbers

Math Matters

QUESTION

The imaginary unit i is important in the field of electrical engineering. However, because the letter i is used by engineers as the symbol for electric current, these engineers use j for the complex unit.

Complex numbers a + bi

Real numbers a + 0i (b = 0)

Imaginary numbers 0 + bi (a = 0, b 0)

What are the real part and imaginary part of 3 5i?

Note from the diagram at the left that the set of real numbers is a subset of the complex numbers, and the set of imaginary numbers is a separate subset of the complex numbers. The set of real numbers and the set of imaginary numbers are disjoint sets. Example 1 illustrates how to write a complex number in the standard form a bi. EXAMPLE 1

»

Write a Complex Number in Standard Form

Write 7 兹45 in the form a bi. 7 兹45 苷 7 i兹45 苷 7 i兹9 兹5 苷 7 3i兹5

Solution

»

Try Exercise 8, page 340

Addition and Subtraction of Complex Numbers All the standard arithmetic operations that are applied to real numbers can be applied to complex numbers. Definition of Addition and Subtraction of Complex Numbers If a bi and c di are complex numbers, then Addition

共a bi兲共c di兲苷共a c兲共b d兲i

Subtraction

共a bi兲共c di兲苷共a c兲共b d兲i

Basically, these definitions say that to add two complex numbers, add the real parts and add the imaginary parts. To subtract two complex numbers, subtract the real parts and subtract the imaginary parts. EXAMPLE 2

»

Add or Subtract Complex Numbers

Simplify. 共7 2i兲共2 4i兲

a.

b.

共9 4i兲共2 6i兲

Solution a.

共7 2i兲共2 4i兲苷共7 共2兲兲共2 4兲i 苷 5 2i

b.

共9 4i兲共2 6i兲苷共9 2兲共4 共6兲兲i 苷 11 10i

»

Try Exercise 18, page 340

ANSWER

Real part: 3; imaginary part: 5

5.1

337

The Real Number System

Multiplication of Complex Numbers When multiplying complex numbers, the term i 2 is frequently a part of the product. Recall that i 2 苷 1. Therefore, 3i共5i兲苷 15i 2 苷 15共1兲苷 15 2i共6i兲苷 12i 2 苷 12共1兲苷 12 4i共3 2i兲苷 12i 8i 2 苷 12i 8共1兲苷 8 12i

take note

When multiplying square roots of negative numbers, first rewrite the radical expressions using i. For instance,

Recall that the definition of the

兹6 兹24 苷 i兹6 i兹24 苷 i 2兹144 苷 1 12 苷 12

product of radical expressions requires that the radicand be a

positive number. Therefore, when

• 兹6 i 兹6, 兹24 苷 i 兹24

negative radicands, we first must

Note from this example that it would have been incorrect to multiply the radicands of the two radical expressions. To illustrate:

rewrite the expression using i and

兹6 兹24 苷兹共6兲共24兲

multiplying expressions containing

a positive radicand. QUESTION

What is the product of 兹2 and 兹8?

To multiply two complex numbers, we use the following definition. Definition of Multiplication of Complex Numbers If a bi and c di are complex numbers, then 共a bi兲共c di兲苷共ac bd兲共ad bc兲i

Because every complex number can be written as a sum of two terms, it is natural to perform multiplication on complex numbers in a manner consistent with the operation defined on binomials and the definition i 2 苷 1. By using this analogy, you can multiply complex numbers without memorizing the definition.

EXAMPLE 3

Simplify.

a.

»

Multiply Complex Numbers

共3 4i兲共2 5i兲

b.

共 2 兹3 兲共 4 5兹3 兲

Solution a.

共3 4i兲共2 5i兲苷 6 15i 8i 20i 2 苷 6 15i 8i 20共1兲苷 6 15i 8i 20 苷 26 7i

• Replace i 2 by 1. • Simplify.

Continued

ANSWER

兹2 兹8 苷 i 兹2 i 兹8 苷 i 2 兹16 苷 1 4 苷 4

䉴

338

Chapter 5

Complex Numbers

Integrating Technology Some graphing calculators can be used to perform operations on complex numbers. Here are some typical screens for a TI-83/TI-83 Plus/TI-84 Plus graphing calculator. Press MODE . Use the down arrow key to highlight a bi.

共 2 兹3 兲共 4 5兹3 兲苷共 2 i兹3 兲共 4 5i兹3 兲

b.

苷 8 10i兹3 4i兹3 5i 2共3兲苷 8 10i兹3 4i兹3 5共1兲共3兲苷 8 10i兹3 4i兹3 15 苷 23 6i兹3

»

Try Exercise 34, page 340

Division of Complex Numbers Recall that the number

3 兹2

is not in simplest form because there is a radical ex-

3 is not in simplest form because i i 苷兹1. To write this expression in simplest form, multiply the numerator and denominator by i.

Normal Sci Eng Float 01 23456789 Radian Degree Func Par Pol Seq Connected Dot Sequential Simul Real a+bi re^θi Full Horiz

pression in the denominator. Similarly,

3 i 3i 3i 苷 2苷苷 3i i i i 1

Press ENTER 2nd [QUIT]. The following screen shows two examples of computations on complex numbers. To enter an i, use 2nd [i], which is located above the decimal point key. (3 –4i) ( 2+5i) 26 +7 i (16 –1 1 i) /( 5+2i) 2–3i

Here is another example. 3 6i 3 6i i 3i 6i 2 3i 6共1兲 3i 6 3 苷苷苷苷苷 3 i 2i 2i i 2i 2 2共1兲 2 2 2 兹3 , we multiply the numerator and de5 2兹3 nominator by the conjugate of 5 2兹3, which is 5 2兹3. In a similar manner, to find the quotient of two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The complex numbers a bi and a bi are called complex conjugates or conjugates of each other. The conjugate of the complex number z is denoted by z. For instance, Recall that to simplify the quotient

2 5i 苷 2 5i

and

3 4i 苷 3 4i

Consider the product of a complex number and its conjugate. For instance, 共2 5i兲共2 5i兲苷 4 10i 10i 25i 2 苷 4 25共1兲苷 4 25 苷 29 Note that the product is a real number. This is always true. Product of Complex Conjugates The product of a complex number and its conjugate is a real number. That is, 共a bi兲共a bi兲苷 a2 b2. Example

共5 3i兲共5 3i兲苷 52 32 苷 25 9 苷 34

5.1

The Real Number System

339

The next example shows how the quotient of two complex numbers is determined by using conjugates.

EXAMPLE 4

Simplify:

»

Divide Complex Numbers

16 11i 5 2i

Solution 16 11i 16 11i 5 2i 苷 5 2i 5 2i 5 2i 80 32i 55i 22i 2 苷 52 22 80 32i 55i 22共1兲苷 25 4 80 87i 22 苷 29 58 87i 苷 29 29共2 3i兲苷苷 2 3i 29

• Multiply numerator and denominator by the conjugate of the denominator.

»

Try Exercise 48, page 341

Powers of i The following powers of i illustrate a pattern: i1 i2 i3 i4

苷i 苷 1 苷 i 2 i 苷共1兲i 苷 i 苷 i 2 i 2 苷共1兲共1兲苷 1

i5 i6 i7 i8

苷 i4 i 苷 1 i 苷 i 苷 i 4 i 2 苷 1共1兲苷 1 苷 i 4 i 3 苷 1共i兲苷 i 苷共i 4 兲2 苷 12 苷 1

Because i 4 苷 1, 共i 4 兲n 苷 1n 苷 1 for any integer n. Thus it is possible to evaluate powers of i by factoring out powers of i 4, as shown in the following: i 27 苷共i 4 兲6 i 3 苷 16 i 3 苷 1 共i兲苷 i The following theorem can also be used to evaluate powers of i.

Powers of i If n is a positive integer, then i n 苷 i r, where r is the remainder of the division of n by 4.

340

Chapter 5

Complex Numbers EXAMPLE 5

»

Evaluate a Power of i

Evaluate: i 153

Solution Use the powers of i theorem. i 153 苷 i 1 苷 i

• Remainder of 153 4 is 1.

»

Try Exercise 60, page 341

Topics for Discussion 1. What is an imaginary number? What is a complex number? 2. How are the real numbers related to the complex numbers? 3. Is zero a complex number? 4. What is the conjugate of a complex number? 5. If a and b are real numbers and ab 苷 0, then a 苷 0 or b 苷 0. Is the same true for complex numbers? That is, if u and v are complex numbers and uv 苷 0, must one of the complex numbers be zero?

Exercise Set 5.1 In Exercises 1 to 10, write the complex number in standard form.

1. 兹81

2. 兹64

3. 兹98

4. 兹27

5. 兹16 兹81

6. 兹25 兹9

7. 5 兹49

8. 6 兹1

9. 8 兹18

10. 11 兹48

19. 8i 共2 8i 兲

20. 3 共4 5i 兲

21. 5i 8i

22. 共3i 兲共2i 兲

23. 兹50 兹2

24. 兹12 兹27

25. 3共2 5i 兲 2共3 2i 兲

26. 3i共2 5i 兲 2i共3 4i 兲

27. 共4 2i 兲共3 4i 兲

28. 共6 5i 兲共2 5i 兲

29. 共3 4i 兲共2 7i 兲

30. 共5 i 兲共2 3i 兲

31. 共4 5i 兲共4 5i 兲

In Exercises 11 to 36, simplify and write the complex number in standard form.

32. 共3 7i 兲共3 7i 兲

11. 共5 2i 兲共6 7i 兲

12. 共4 8i 兲共5 3i 兲

33. 共 3 兹4 兲共 2 兹9 兲

13. 共2 4i 兲共5 8i 兲

14. 共3 5i 兲共8 2i 兲

34. 共 5 2兹16 兲共 1 兹25 兲

15. 共1 3i 兲共7 2i 兲

16. 共2 6i 兲共4 7i 兲

35. 共 3 2兹18 兲共 2 2兹50 兲

17. 共3 5i 兲共7 5i 兲

18. 共5 3i 兲共2 9i 兲

36. 共 5 3兹48 兲共 2 4兹27 兲

5.1 In Exercises 37 to 54, write each expression as a complex number in standard form.

37.

6 i

38.

8 2i

39.

6 3i i

40.

4 8i 4i

41.

1 7 2i

42.

5 3 4i

43.

2i 1i

44.

5i 2 3i

45.

5i 4 5i

46.

4i 3 5i

47.

3 2i 3 2i

48.

8i 2 3i

7 26i 4 3i

50.

49.

52. 共2 4i 兲2

53. 共1 2i 兲3

54. 共2 i 兲3

341

FRACTAL GEOMETRY Many fractal images are generated by substituting an initial value into a complex function, calculating the output, and then using the output as the next value to substitute into the function. The process is then repeated indefinitely. This recycling of outputs is called iteration, and each output is called an iterate. In Exercises 63 to 66, we will use the letter z to symbolize a complex number. The initial value is called the seed and is denoted by z0. Successive iterates are denoted by z1, z2, z3, … .

63. Let f共z兲苷 z 4 3i. Begin with the intial value z0 苷 2 i. Determine z1, z2, …, z5. 64. Let f共z兲苷 2iz. Begin with the intial value z0 苷 1 3i. Determine z1, z2, …, z5. 65. Let f共z兲苷 iz. Begin with the intial value z0 苷 1 i. Determine z1, z2, z3, and z4. Find a pattern and use it to predict z5, z6, z7, and z8. 66. Let f共z兲苷 z 2. Begin with the intial value z0 苷 0.5i. Determine z1, z2, and z3.

4 39i 5 2i

51. 共3 5i 兲2

The Real Number System

b 兹b 2 4ac for the 2a given values of a, b, and c. Write your answer as a complex number in standard form. In Exercises 67 to 72, evaluate

67. a 苷 3, b 苷 3, c 苷 3

68. a 苷 2, b 苷 4, c 苷 4

69. a 苷 2, b 苷 6, c 苷 6

70. a 苷 2, b 苷 1, c 苷 3

71. a 苷 4, b 苷 4, c 苷 2

72. a 苷 3, b 苷 2, c 苷 4

In Exercises 55 to 62, evaluate the power of i.

55. i 15 59.

56. i 66

1 i 25

60.

1 i 83

57. i 40

58. i 51

61. i 34

62. i 52

Connecting Concepts The property that the product of conjugates of the form (a bi )(a bi ) is equal to a 2 b 2 can be used to factor the sum of two perfect squares over the set of complex numbers. For example, x 2 y 2 (x yi )(x yi ). In Exercises 73 to 78, factor the binomial over the set of complex numbers.

73. x 2 16

74. x 2 9

75. z 2 25

76. z 2 64

77. 4x 2 81

78. 9x 2 1

79. Show that if x 苷 1 2i, then x 2 2x 5 苷 0. 80. Show that if x 苷 1 2i, then x 2 2x 5 苷 0. 3

81. When we think of the cube root of 8, 兹 8, we normally 3 mean the real cube root of 8 and write 兹 8 苷 2. However, there are two other cube roots of 8 that are complex numbers. Verify that 1 i 兹 3 and 1 i 兹 3 are 3 cube roots of 8 by showing that 共 1 i 兹 3 兲苷 8 and 3 共 1 i 兹 3 兲苷 8.

342

Chapter 5

Complex Numbers

82. It is possible to find the square root of a complex number. 兹2 Verify that 兹 i 苷共1 i 兲 by showing that 2

冋

83. Simplify i i 2 i 3 i 4 i 28. 84. Simplify i i 2 i 3 i 4 i 100.

册

2 兹2 共1 i兲苷 i. 2

Projects ARGAND DIAGRAM Just as we can graph a real number on a real number line, we can graph a complex number. This is accomplished by using one number line for the real part of the complex number and one number line for the imaginary part of the complex number. These two number lines are drawn perpendicular to each other and pass through their respective origins, as shown below. The result is called the complex plane or an Argand diagram after Jean-Robert Argand (1768–1822), an accountant and amateur mathematician. Although he is given credit for this representation of complex numbers, Caspar Wessel (1745–1818) actually conceived the idea before Argand. To graph the complex number 3 4i, start at 3 on the real axis. Now move 4 units up (for positive numbers move up; for negative numbers move down) and place a dot at that point, as shown in the diagram. The graphs of several other complex numbers are also shown. Imaginary axis

8

− 5 + 7i

4

3

3 + 4i 4

−8

−4

4 −4

− 4 − 6i −8

2 − 5i

8

Real axis

In Exercises 1 to 8, graph the complex number.

1. 2 5i

2. 4 3i

3. 2 6i

4. 3 5i

5. 4

6. 2i

7. 3i

8. 5

The absolute value of a complex number is given by |a bi | 兹a 2 b 2 . In Exercises 9 to 12, find the absolute value of the complex number.

9. 2 5i

10. 4 3i

11. 2 6i

12. 3 5i

13. The additive inverse of a bi is a bi. Show that the absolute value of a complex number and the absolute value of its additive inverse are equal. 14. A real number and its additive inverse are the same distance from zero but on opposite sides of zero on a real number line. Describe the relationship between the graphs of a complex number and its additive inverse.

5.2

Section 5.2 ■

■

Trigonometric Form of a Complex Number The Product and Quotient of Complex Numbers Written in Trigonometric Form

Trigonometric Form of Complex Numbers

343

Trigonometric Form of Complex Numbers P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A22.

PS1. Simplify: 共1 i兲共2 i兲 [5.1] PS2. Simplify:

2i [5.1] 3i

PS3. What is the conjugate of 2 3i ? [5.1] PS4. What is the conjugate of 3 5i ? [5.1] PS5. Use the quadratic formula to find the solutions of x 2 x 苷 1. [1.1/5.1] PS6. Solve: x 2 9 苷 0 [1.1/5.1]

Trigonometric Form of a Complex Number Real numbers are graphed as points on a number line. Complex numbers can be graphed in a coordinate plane called the complex plane. The horizontal axis of the complex plane is called the real axis; the vertical axis is called the imaginary axis.

A complex number written in the form z 苷 a bi is written in standard form or rectangular form. The graph of a bi is associated with the point P共a, b兲 in the complex plane. Figure 5.1 shows the graphs of several complex numbers. The length of the line segment from the origin to the point 共3, 4兲 in the complex plane is the absolute value of z 苷 3 4i. See Figure 5.2. From the Pythagorean Theorem, the absolute value of z 苷 3 4i is 兹共3兲2 42 苷兹25 苷 5 Im

4

Imaginary axis z = −3 + 4i

2 + 4i

−2 + 3i

Im

5

Real axis

−4

4

|z| = 5

Re

−3 − 3i −4

−4i

Figure 5.1

−4

4 Re

Figure 5.2

344

Chapter 5

Complex Numbers

Im

Definition of the Absolute Value of a Complex Number z = a + bi

The absolute value of the complex number z 苷 a bi, denoted by 兩z兩, is

|z| = a 2 + b 2

兩z兩苷兩a bi兩苷兹a2 b 2

b

a

Figure 5.3

Re

Thus 兩z兩 is the distance from the origin to z (see Figure 5.3). QUESTION

The conjugate of a bi is a bi. Does 兩a bi兩苷兩a bi兩?

A complex number z 苷 a bi can be written in terms of trigonometric functions. Consider the complex number graphed in Figure 5.4. We can write a and b in terms of the sine and the cosine. a cos 苷 r a 苷 r cos

Im

P(a, b)

|z| = r

b sin 苷 r b 苷 r sin

b = r sin θ

θ a = r cos θ

Re

Figure 5.4

where r 苷兩z兩苷兹a2 b 2. Substituting for a and b in z 苷 a bi, we obtain z 苷 r cos u ir sin u 苷 r共cos u i sin u兲

The expression z 苷 r共cos u i sin u兲 is known as the trigonometric form of a complex number. The trigonometric form of a complex number is also called the polar form of the complex number. The notation cos u i sin u is often abbreviated as cis using the c from cos , the imaginary unit i, and the s from sin u. Trigonometric Form of a Complex Number The complex number z 苷 a bi can be written in trigonometric form as z 苷 r共cos i sin 兲苷 r cis where a 苷 r cos , b 苷 r sin , r 苷兹a2 b 2, and tan 苷

b . a

In this text we will often write the trigonometric form of a complex number in its abbreviated form z 苷 r cis u. The value of r is called the modulus of the complex number z, and the angle u is called the argument of the complex number z. The modulus r and the argument u of a complex number z 苷 a bi are given by r 苷兹a2 b 2

ANSWER

and

cos 苷

Yes, because 兹a2 b2 苷兹a2 共b兲2.

a , r

sin 苷

b r

5.2

345

Trigonometric Form of Complex Numbers

兩兩

b , where is the reference angle for . Due to a the periodic nature of the sine and cosine functions, the trigonometric form of a complex number is not unique. Because cos 苷 cos共 2k兲 and sin 苷 sin共 2k兲, where k is an integer, the complex numbers We also can write 苷 tan1

r cis u and r cis共u 2kp兲 are equal. For example, 2 cis EXAMPLE 1

»

冉

冊

苷 2 cis 2 . 6 6

Write a Complex Number in Trigonometric Form

Write z 苷 2 2i in trigonometric form.

Solution Find the modulus and the argument of z. Then substitute these values in the trigonometric form of z.

Im

r 苷兹共2兲2 共2兲2 苷兹8 苷 2兹2

2

To determine u, we first determine a. See Figure 5.5.

θ −2

α

苷 tan1

Re

2

r

z = −2 − 2i

−2

a 苷 tan1

兩兩兩兩 b a

• is the reference angle of angle .

2 苷 tan1 1 苷 45 2

• a 2 and b 2

苷 180 45 苷 225

Figure 5.5

• Because z is in the third quadrant, 180° v v 270°.

The trigonometric form is z 苷 r cis u 苷 2兹2 cis 225

• r 2兹2, 225

»

Try Exercise 12, page 349

EXAMPLE 2 Im

−1 + i 3

2

r=2

»

Write a Complex Number in Standard Form

Write z 苷 2 cis 120 in standard form.

Solution

120°

−2

2 −1

Figure 5.6

Re

Write z in the form r共cos i sin 兲 and then evaluate cos and sin . See Figure 5.6.

冉

z 苷 2 cis 120 苷 2共cos 120 i sin 120 兲苷 2

»

Try Exercise 30, page 349

冊

1 兹3 i 苷 1 i 兹3 2 2

346

Chapter 5

Complex Numbers

The Product and Quotient of Complex Numbers Written in Trigonometric Form Let z1 and z2 be two complex numbers written in trigonometric form. The product of z1 and z2 can be found by using trigonometric identities. If z1 苷 r1共cos 1 i sin 1兲 and z2 苷 r2共cos 2 i sin 2兲, then z1 z 2 苷 r1共cos u1 i sin u1 兲 r2共cos u2 i sin u2 兲苷 r1 r2共cos u1 cos u2 i cos u1 sin u2 i sin u1 cos u2 i 2 sin u1 sin u2 兲苷 r1 r2 关共cos u1 cos u2 sin u1 sin u2 兲 i共sin u1 cos u2 cos u1 sin u2 兲兴苷 r1 r2 关cos共u1 u2 兲 i sin共u1 u2 兲兴 • Identities for cos( 1 2) and sin( 1 2)

z1 z 2 苷 r1 r2 cis共u1 u2 兲.

Thus, using cis notation, we have

The above result is called the Product Property of Complex Numbers. It states that the modulus for the product of two complex numbers in trigonometric form is the product of the moduli of the two numbers, and the argument of the product is the sum of the arguments of the two numbers.

EXAMPLE 3

2

1 2

−2

Solution

2

z2 = − 3 + i

Find the Product of Two Complex Numbers

Find the product of z1 苷 1 i兹3 and z2 苷兹3 i by using the trigonometric forms of the complex numbers. Write the answer in standard form.

Im

z1 = −1 + i 3

»

5π 6

−1

2π 3

Write each complex number in trigonometric form. Then use the Product Property of Complex Numbers. See Figure 5.7. 1

2p 3 5p z2 苷兹3 i 苷 2 cis 6 2 5 z1z 2 苷 2 cis 2 cis 3 6 2 5 苷 4 cis 3 6 3 苷 4 cis 2 3 3 苷 4 cos i sin 2 2 苷 4共0 i兲苷 4i z1 苷 1 i兹3 苷 2 cis

Re

Figure 5.7

冉

冉

»

Try Exercise 56, page 349

冊

冊

• r1 2, 1

2 3

• r 2 2, 2

5 6

• The Product Property of Complex Numbers • Simplify.

5.2

Trigonometric Form of Complex Numbers

347

Similarly, the quotient of z1 and z2 can be found by using trigonometric identities. If z1 苷 r1共cos u1 i sin u1 兲 and z2 苷 r2共cos u2 i sin u2 兲, then z1 r1共cos 1 苷 z2 r2共cos 2 r1共cos 1 苷 r2共cos 2 苷

i sin 1兲 i sin 2兲 i sin 1兲共cos 2 i sin 2兲 i sin 2兲共cos 2 i sin 2兲

r1共cos 1 cos 2 i cos 1 sin 2 i sin 1 cos 2 i 2 sin 1 sin 2兲 r2共cos2 2 i 2 sin2 2兲

r1关共cos 1 cos 2 sin 1 sin 2兲 i共sin 1 cos 2 cos 1 sin 2兲兴 r2共cos2 2 sin2 2兲 r1 • Identities for cos ( 1 2), 苷关cos共1 2兲 i sin共1 2兲兴 r2 sin ( 1 2), and

苷

cos2 2 sin2 2

Thus, using cis notation, we have

z1 r1 苷 cis共1 2兲. z 2 r2

The above result is called the Quotient Property of Complex Numbers. It states that the modulus for the quotient of two complex numbers in trigonometric form is the quotient of the moduli of the two numbers, and the argument of the quotient is the difference of the arguments of the two numbers.

EXAMPLE 4

1

θ1 −1

Find the Quotient of Two Complex Numbers

Find the quotient of z1 苷 1 i and z2 苷兹3 i by using the trigonometric forms of the complex numbers. Write the answer in standard form. Round approximate constants to the nearest thousandth.

Im

z1 = −1 + i

»

Solution 1

2

θ2 −1

z2 =

Re

Write the numbers in trigonometric form. Then use the Quotient Property of Complex Numbers. See Figure 5.8.

3−i

Figure 5.8

z1 苷 1 i 苷兹2 cis 135

• r1 兹2, 1 135

z2 苷兹3 i 苷 2 cis 330

• r2 2, 2 330°

z1 1 i 兹2 cis 135 苷苷 z 2 兹3 i 2 cis 330 兹2 苷 cis共195 兲 2 苷

兹2 关cos共195 兲 i sin共195 兲兴 2

苷

兹2 共cos 195 i sin 195 兲 2

⬇ 0.683 0.183i

»

Try Exercise 60, page 349

• The Quotient Property of Complex Numbers • Simplify.

• cos(x) cos x, sin(x) sin x

348

Chapter 5

Complex Numbers Here is a summary of the product and quotient theorems.

Product and Quotient Properties of Complex Numbers Written in Trigonometric Form

Product Property of Complex Numbers Let z1 苷 r1共cos u1 i sin u1 兲 and z2 苷 r2共cos u2 i sin u2 兲. Then z1 z 2 苷 r1 r2关cos共1 2兲 i sin共1 2兲兴 z1 z 2 苷 r1 r2 cis共1 2兲

• Using cis notation

Quotient Property of Complex Numbers Let z1 苷 r1共cos u1 i sin u1 兲 and z2 苷 r2共cos u2 i sin u2 兲. Then z1 r1 苷关cos共1 2兲 i sin共1 2兲兴 z 2 r2 z1 r1 苷 cis共1 2兲 • Using cis notation z 2 r2

Topics for Discussion 1. Describe an algebraic procedure that can be used to verify that the absolute value of a bi is equal to the absolute value of a bi. 2. Describe the graph of all complex numbers with an absolute value of 5. 3. Describe two different methods that can be used to simplify

4 . i

4. The complex numbers z1 and z2 both have an absolute value of 1. What is the absolute value of the product z1 z2 ? Explain. 5. Explain how to use the Product Property of Complex Numbers to prove that the product of a complex number and its conjugate is a real number.

5.2

349

Trigonometric Form of Complex Numbers

Exercise Set 5.2 In Exercises 1 to 8, graph each complex number. Find the absolute value of each complex number.

In Exercises 39 to 46, multiply the complex numbers. Write the answer in trigonometric form.

1. z 苷 2 2i

2. z 苷 4 4i

3. z 苷兹3 i

39. 2 cis 30 3 cis 225

4. z 苷 1 i 兹3

5. z 苷 2i

6. z 苷 5

41. 3共cos 122 i sin 122 兲 4共cos 213 i sin 213 兲

7. z 苷 3 5i

8. z 苷 5 4i

42. 8共cos 88 i sin 88 兲 12共cos 112 i sin 112 兲

In Exercises 9 to 20, write each complex number in trigonometric form.

9. z 苷 1 i

10. z 苷 4 4i

11. z 苷兹3 i

12. z 苷 1 i 兹3

13. z 苷 3i

14. z 苷 2i

15. z 苷 5

16. z 苷 3

冉

43. 5 cos

44. 5 cis

18. z 苷 2兹2 2i兹2

19. z 苷 2 2i兹3

20. z 苷兹2 i兹2

In Exercises 21 to 38, write each complex number in standard form.

21. z 苷 2共cos 45 i sin 45 兲 22. z 苷 3共cos 240 i sin 240 兲

32 cis 30 4 cis 150

48.

15 cis 240 3 cis 135

49.

27共cos 315 i sin 315 兲 9共cos 225 i sin 225 兲

50.

9共cos 25 i sin 25 兲 3共cos 175 i sin 175 兲

26. z 苷 cis 315 28. z 苷 5 cis 90

冊冊冊

5 5 i sin 6 6

30. z 苷 4 cos

5 5 i sin 3 3

31. z 苷 3 cos

3 3 i sin 2 2

25共cos 3.5 i sin 3.5兲 5共cos 1.5 i sin 1.5兲

3 52. 5 cis 4 10 cis

54.

18共cos 0.56 i sin 0.56兲 6共cos 1.22 i sin 1.22兲

In Exercises 55 to 62, perform the indicated operation in trigonometric form. Write the solution in standard form. Round approximate constants to the nearest ten-thousandth.

32. z 苷 5共cos i sin 兲

33. z 苷 8 cis

3 4

34. z 苷 9 cis

35. z 苷 9 cis

11 6

36. z 苷 cis

37. z 苷 2 cis 2

45. 4 cis 2.4 6 cis 4.1

47.

53.

冉冉冉

冊

In Exercises 47 to 54, divide the complex numbers. Write the answer in standard form. Round approximate constants to the nearest thousandth.

2 3 51. 11 4 cis 6

24. z 苷 5共cos 120 i sin 120 兲

29. z 苷 2 cos

11 4 3 cis 12 3

12 cis

23. z 苷共cos 315 i sin 315 兲

27. z 苷 8 cis 0

冊冉

2 2 2 2 i sin 2 cos i sin 3 3 5 5

46. 7 cis 0.88 5 cis 1.32

17. z 苷 8 8i兹3

25. z 苷 6 cis 135

40. 4 cis 120 6 cis 315

4 3

3 2

38. z 苷 5 cis 4

55. 共 1 i 兹3 兲共1 i兲

56. 共兹3 i 兲共 1 i 兹3 兲

57. 共3 3i兲共1 i兲

58. 共2 2i兲共兹3 i 兲

59.

1 i 兹3 1 i 兹3

60.

1i 1i

61.

兹2 i 兹2 1i

62.

1 i 兹3 4 4i

350

Chapter 5

Complex Numbers

Connecting Concepts In Exercises 63 to 68, perform the indicated operation in trigonometric form. Write the solution in standard form.

67. 共1 3i兲共2 3i兲共4 5i兲

63. 共兹3 i 兲共2 2i兲共 2 2i 兹3 兲

兹3 i 兹3

共 1 i 兹3 兲共2 2i兲

共2 5i兲共1 6i兲 3 4i

In Exercises 69 and 70, the notation z is used to represent the conjugate of the complex number z.

64. 共1 i兲共 1 i 兹3 兲共兹3 i 兲 65.

68.

66.

共 2 2i 兹3 兲共 1 i 兹3 兲

69. Use the trigonometric forms of z and z to find z z.

4 兹3 4i

70. Use the trigonometric forms of z and z to find

z z

.

Projects number by i, the imaginary unit. Now multiply a complex number by i and note the effect. The use of a complex plane may be helpful in your explanation.

1. A GEOMETRIC INTERPRETATION Multiplying a real number by a number greater than 1 increases the magnitude of that number. For example, multiplying 2 by 3 triples the magnitude of 2. Explain the effect of multiplying a real

Section 5.3 ■ ■

De Moivre’s Theorem De Moivre’s Theorem for Finding Roots

De Moivre’s Theorem P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A22.

PS1. Find

冉

冊

2

兹2 兹2 i . [5.1] 2 2

PS2. Find the real root of x 3 8 苷 0. PS3. Find the real root of x 5 243 苷 0. PS4. Write 2 2i in trigonometric form. [5.2] PS5. Write 2共cos 150 i sin 150 兲 in standard form. [5.2] PS6. Find the absolute value of

兹2 兹2 i. [5.2] 2 2

De Moivre’s Theorem De Moivre’s Theorem is a procedure for finding powers and roots of complex numbers when the complex numbers are expressed in trigonometric form. This theorem can be illustrated by repeated multiplication of a complex number. Let z 苷 r cis u. Then z2 can be written as z z 苷 r cis r cis z2 苷 r 2 cis 2

5.3

Math Matters

De Moivre’s Theorem

351

The product z2 z is z2 z 苷 r 2 cis 2 r cis z3 苷 r 3 cis 3 If we continue this process, the results suggest a formula known as De Moivre’s Theorem for the nth power of a complex number. De Moivre’s Theorem If z 苷 r cis and n is a positive integer, then zn 苷 r n cis n

EXAMPLE 1 Abraham de Moivre (1667–1754) was a French mathematician who fled to England during the expulsion of the Huguenots in 1685. De Moivre made important contributions to probability theory and analytic geometry. In 1718 he published The Doctrine of Chance, in which he developed the theory of annuities, mortality statistics, and the concept of statistical independence. In 1730 de Moivre stated the theorem shown above Example 1, which we now call De Moivre’s Theorem. This theorem is significant because it provides a connection between trigonometry and mathematical analysis. Although de Moivre was well respected in the mathematical community and was elected to the Royal Society of England, he never was able to secure a university teaching position. His income came mainly from tutoring, and he died in poverty. De Moivre is often remembered for predicting the day of his own death. At one point in his life, he noticed that he was sleeping a few minutes longer each night. Thus he calculated that he would die when he needed 24 hours of sleep. As it turned out, his calculation was correct.

»

Find the Power of a Complex Number

Find 共2 cis 30 兲5. Write the answer in standard form.

Solution By De Moivre’s Theorem, 共2 cis 30 兲5 苷 25 cis共5 30 兲苷 25关cos共5 30 兲 i sin共5 30 兲兴苷 32共cos 150 i sin 150 兲

冉

苷 32

冊

1 兹3 i 苷 16兹3 16i 2 2

»

Try Exercise 6, page 354

EXAMPLE 2

»

Use De Moivre’s Theorem

Find 共1 i兲8 using De Moivre’s Theorem. Write the answer in standard form.

Solution Convert 1 i to trigonometric form and then use De Moivre’s Theorem. 共1 i兲8 苷共兹2 cis 45 兲8 苷共兹2 兲8 cis 8共45 兲苷 16 cis 360 苷 16共cos 360 i sin 360 兲苷 16共1 0i兲苷 16

»

Try Exercise 16, page 354

QUESTION

ANSWER

Is 共1 i兲4 a real number?

Yes. 共1 i兲4 苷共兹2 cis 45 兲苷共兹2 兲 cis 180 苷 4. 4

4

352

Chapter 5

Complex Numbers

De Moivre’s Theorem for Finding Roots De Moivre’s Theorem can be used to find the nth roots of any number. De Moivre’s Theorem for Finding Roots If z 苷 r cis is a complex number, then there exist n distinct nth roots of z given by wk 苷 r 1/n cis

EXAMPLE 3

»

360 k for k 苷 0, 1, 2, . . . , n 1, and n 1 n

Find Cube Roots by De Moivre’s Theorem

Find the three cube roots of 27.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

Write 27 in trigonometric form: 27 苷 27 cis 0 . Then, from De Moivre’s Theorem for finding roots, the cube roots of 27 are wk 苷 27 1/3 cis

Im

4 −

0 360 k for k 苷 0, 1, 2 3

3 3 3 + i 2 2

Substitute for k to find the three cube roots of 27. w0 苷 27 1/3 cis 0

• k 0;

0 360 (0) 0 3

苷 3共cos 0 i sin 0 兲苷3 w1 苷 27

1/3

cis 120

0 360 (1) • k 1; 120 3

• k 2;

0 360 (2) 240 3

苷 3共cos 240 i sin 240 兲 3 3兹3 苷 i 2 2 0 1080 苷 360 . The angles start repeating; thus there are 3 only three cube roots of 27. The three cube roots are graphed in Figure 5.9. For k 苷 3,

»

Try Exercise 28, page 355

4 Re

−

苷 3共cos 120 i sin 120 兲 3 3兹3 苷 i 2 2 w2 苷 27 1/3 cis 240

3 −4

3 3 3 − i 2 2

−4

Figure 5.9

Note that the arguments of the three cube roots of 27 are 0°, 120°, and 240° and that 兩w0兩苷兩w1兩苷兩w2兩苷 3. In geometric terms, this means that the three cube roots of 27 are equally spaced on a circle centered at the origin with a radius of 3.

5.3

EXAMPLE 4

»

353

De Moivre’s Theorem

Find the Fifth Roots of a Complex Number

Find the fifth roots of z 苷 1 i兹3.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

Write z in trigonometric form: z 苷 r cis .

Im

r 苷兹12 共兹3 兲2 苷 2 z 苷 2 cis 60

●

tan1

兹3 60 1

5 From De Moivre’s Theorem, the modulus of each root is 兹 2, and the 60 360 k arguments are determined by , k 苷 0, 1, 2, 3, 4. 5 5 wk 苷兹 2 cis

60 360 k 5

●

k 0, 1, 2, 3, 4

Substitute for k to find the five fifth roots of z. 5 w0 苷兹 2 cis 12

• k 0;

60 360 (0) 12 5

5 w1 苷兹 2 cis 84

• k 1;

60 360 (1) 84 5

5 w2 苷兹 2 cis 156

• k 2;

60 360 (2) 156 5

5 w3 苷兹 2 cis 228

• k 3;

60 360 (3) 228 5

5 w4 苷兹 2 cis 300

• k 4;

60 360 (4) 300 5

1

r=

5

2

−1

1

Re

−1

Figure 5.10

The five fifth roots of 1 i 兹3 are graphed in Figure 5.10. Note that the roots are equally spaced on a circle with center 共0, 0兲 and a 5 radius of 兹 2 ⬇ 1.15.

»

Try Exercise 30, page 355

Keep the following properties in mind as you compute the n distinct nth roots of the complex number z. Properties of the n th Roots of z

Geometric Property

Integrating Technology A web applet is available to show the nth roots of a complex number. This applet, Nth Roots, can be found on our website at college.hmco.com/info/ aufmannCAT

All nth roots of z are equally spaced on a circle with center 共0, 0兲 and a radius of 兩z兩1/n.

Absolute Value Properties 1. If 兩z兩苷 1, then each nth root of z has an absolute value of 1. 2. If 兩z兩 1, then each nth root of z has an absolute value of 兩z兩1/n, where 兩z兩1/n is greater than 1 but less than 兩z兩. 3. If 兩z兩 1, then each nth root of z has an absolute value of 兩z兩1/n, where 兩z兩1/n is less than 1 but greater than 兩z兩.

354

Chapter 5

Complex Numbers

Argument Property and the n arguments of the remaining nth roots can be determined by adding 360 2 multiples of (or if you are using radians) to . n n n Given that the argument of z is , then the argument of w0 is

QUESTION

Are all fourth roots of 1 equally spaced on a circle with center 共0, 0兲 and radius 1?

Topics for Discussion 1. How many solutions are there for z 苷共a bi兲8? How many solutions are there for z8 苷 a bi? Explain. 2. To solve z4 苷 16, a student first observes that z 苷 2 is one solution. The other solutions are equally spaced on a circle with center 共0, 0兲 and radius 2, so the student reasons that z 苷 2i, z 苷 2, and z 苷 2i are the other three solutions. Do you agree? Explain. 3. If 兩z兩苷 1, then the n solutions of w n 苷 z all have an absolute value of 1. Do you agree? Explain. 4. If z is a solution of z2 苷 c di, then the conjugate of z is also a solution. Do you agree? Explain. ANSWER

Yes.

Exercise Set 5.3 In Exercises 1 to 16, find the indicated power. Write the answer in standard form.

1. 关2共cos 30 i sin 30 兲兴 8

2. 共cos 240 i sin 240 兲12

3. 关2共cos 240 i sin 240 兲兴 5 4. 关2共cos 45 i sin 45 兲兴 10 5. 共2 cis 225 兲5 7.

冉

2 cis

冊

2 3

9. 共1 i兲10 11. 共1 i兲4

6. 共2 cis 330 兲4

6

8.

冉

4 cis

冊

5 6

3

15.

冉

冊

兹2 兹2 i 2 2

6

16.

冉

冊

兹2 兹2 i 2 2

12

In Exercises 17 to 30, find all of the indicated roots. Write all answers in standard form. Round approximate constants to the nearest thousandth.

17. The two square roots of 9 18. The two square roots of 16

10. 共 1 i 兹3 兲

19. The six sixth roots of 64

12. 共 2 2i 兹3 兲3

20. The five fifth roots of 32

8

14. 共 2 兹 3 2i 兲

5

13. 共2 2i兲7

5.3 21. The five fifth roots of 1 22. The four fourth roots of 16

De Moivre’s Theorem

355

In Exercises 31 to 42, find all roots of the equation. Write the answers in trigonometric form.

31. x 3 8 苷 0

32. x 5 32 苷 0

23. The three cube roots of 1

33. x 4 i 苷 0

34. x 3 2i 苷 0

24. The three cube roots of i

35. x 3 27 苷 0

36. x 5 32i 苷 0

37. x 4 81 苷 0

38. x 3 64i 苷 0

26. The five fifth roots of 1 i

39. x 4 共 1 i 兹3 兲苷 0

40. x 3 共 2 兹3 2i 兲苷 0

27. The three cube roots of 2 2i 兹3

41. x 3 共 1 i 兹3 兲苷 0

42. x 6 共4 4i兲苷 0

25. The four fourth roots of 1 i

28. The three cube roots of 2 2i 兹3 29. The two square roots of 16 16i 兹3 30. The two square roots of 1 i 兹3

Connecting concepts

Connecting Concepts 43. Show that the conjugate of z 苷 r共cos i sin 兲 is equal to z 苷 r共cos i sin 兲. 44. Show that if z 苷 r共cos i sin 兲, then z 1 苷 r 1共cos i sin 兲 45. Show that if z 苷 r共cos i sin 兲, then z 2 苷 r 2共cos 2 i sin 2兲 Note that Exercises 44 and 45 suggest that if z r(cos i sin ), then, for positive integers n,

z n r n(cos n i sin n )

47. SUM OF THE nTH ROOTS OF 1 Make a conjecture about the sum of the nth roots of 1 for any natural number n 2. (Hint: Experiment by finding the sum of the two square roots of 1, the sum of the three cube roots of 1, the sum of the four fourth roots of 1, the sum of the five fifth roots of 1, and the sum of the six sixth roots of 1.) 48. PRODUCT OF THE nTH ROOTS OF 1 Make a conjecture about the product of the nth roots of 1 for any natural number n 2. (Hint: Experiment by finding the product of the two square roots of 1, the product of the three cube roots of 1, the product of the four fourth roots of 1, the product of the five fifth roots of 1, and the product of the six sixth roots of 1.)

46. Use the above equation to find z4 for z 苷 1 i 兹3.

Projects 1. VERIFY IDENTITIES Raise 共cos i sin 兲 to the second power by using De Moivre’s Theorem. Now square 共cos i sin 兲 as a binomial. Equate the real and imaginary parts of the two complex numbers and show that

2. DISCOVER IDENTITIES Raise 共cos i sin 兲 to the fourth power by using De Moivre’s Theorem. Now find the fourth power of the binomial 共cos i sin 兲 by multiplying. Equate the real and imaginary parts of the two complex numbers.

a. cos 2 苷 cos2 sin2

a. What identity have you discovered for cos 4?

b. sin 2 苷 2 sin cos

b. What identity have you discovered for sin 4?

356

Chapter 5

Complex Numbers

Exploring Concepts with Technology The Mandelbrot Iteration Procedure The following procedure is called the Mandelbrot iteration procedure. Mandelbrot Iteration Procedure Choose a complex number z0. 1. Square z0 and add the reult to z0. 2. Square the preceding result and add it to z0. 3. Repeat step 2.

Applying the Mandelbrot iteration procedure to a complex number z0 generates a sequence of numbers z1, z2, z3, … . The numbers z1, z2, z3, … are called iterates. In this case, the iterates are generated using the complex function f共z兲苷 z2 z0. The number z0 is referred to as the seed or initial value. If you change the seed, a different sequence of iterates is generated. Some seeds generate iterates that grow without bound; some seeds generate iterates that approach a constant; and some seeds generate iterates that are cyclic. For instance: ■

Let z0 苷 1. Then z1 苷 12 1 苷 2, z2 苷 22 1 苷 5, z3 苷 52 1 苷 26, z4 苷 262 1 苷 677 In this case, the iterates grow larger and larger.

■

Let z0 苷 1. Then z1 苷 (1)2 (1) 苷 0, z2 苷 02 (1) 苷 1, z3 苷 (1)2 (1) 苷 0 In this case, the iterates cycle: 0, 1, 0, 1, 0, … .

■

Let z0 苷 0.25. Then z1 苷 0.252 0.25 苷 0.3125, z2 苷 0.31252 0.25 苷 0.34765625, z3 苷 0.347656252 0.25 ⬇ 0.3708648682 In this case, subsequent iterates can be calculated easily using a graphing calculator and the method shown below. Store the seed in Z. Press ENTER. 0.25→Z Store Z2 plus the seed in Z. Press Z2+0.25→Z ENTER to produce the next iterate.

.25 = Z0 .3125 .34765625 .3708648682 .3875407504

= Z1 = Z2 = Z3 = Z4

1. Use a graphing calculator to continue the above Mandelbrot iteration procedure, with z0 苷 0.25, to find z5, z10, z100, and z200. It can be shown that the sequence of iterates is approaching a constant. What constant do you think the sequence of iterates is approaching?

Summary

357

2. Let z0 苷 i. Use the Mandelbrot iterartion procedure to find z1, z2, z3, and z4. What happens as the iteration procedure progresses? The black region in Figure 5.11 is called the Mandelbrot set. The Mandelbrot set consists of all complex numbers for which the absolute value of each of their Mandelbrot iteration procedure iterates z1, z2, z3, ... is less than or equal to 2. 1.25

Imaginary part

0.75 0.25 0.00 –0.25 –0.75 –1.25 – 2.00

–1.50

–1.00

–0.50

0.00

0.50

Real part

Figure 5.11

The Mandelbrot set has been called the most complex object in the realm of mathematics. There are many Internet sites that provide additional information on the Mandelbrot set. Visit some of these sites and use the computer programs that are provided to graph the Mandelbrot set and zoom in on different regions of the graph to see some of its complex structure.

Chapter 5 Summary 5.1 Complex Numbers

5.2 Trignometric Form of Complex Numbers

• The number i, called the imaginary unit, is the number such that i2 苷 1.

• The complex number z 苷 a bi can be written in trigonometric form as

• If a is positive real number, then 兹a 苷 i兹a. The number i兹a is called an imaginary number. • A complex number is a number of the form a bi, where a and b are real numbers and i 苷兹1. The number a is the real part of a bi, and b is the imaginary part. • The complex number a bi and a bi are called complex conjugates or conjugates of each other. • Operations on Complex Numbers (a bi) (c di) 苷 (a c) (b d)i (a bi) (c di) 苷 (a c) (b d)i

z 苷 r(cos u i sin u) 苷 r cis u where a 苷 r cos u, b 苷 r sin u, r 苷兹a2 b2, and tan u 苷

• If z1 苷 r1(cos u1 i sin u1) and z2 苷 r2(cos u2 i sin u2), then z1z 2 苷 r1r2 cis(u1 u2)

and

r1 z1 苷 cis(u1 u2) z 2 r2

5.3 De Moivre’s Theorem • De Moivre’s Theorem If z 苷 r cis u and n is a positive integer, then z n 苷 r n cis nu

(a bi)(c di) 苷 (ac bd) (ad bc)i a bi a bi c di 苷 c di c di c di

• If z 苷 r cis u, then the n distinct roots of z are given by • Multiply the numerator and denominator by the conjugate of the denominator.

b a

wk 苷 r 1兾n cis

u 360°k for k 苷 0, 1, 2, … , n 1 n

358

Chapter 5

Complex Numbers

Chapter 5 Assessing Concepts 1. True or false: The real number 7 is also a complex number.

6. How many solutions exist for the equation z 苷 (1 i)5?

2. True or false: The product of a complex number z and its conjugate z– is a real number.

7. How many solutions exist for the equation z5 苷 (1 i)?

3. True or false: z 苷 cos 45 i sin 45 is a square root of i. 4. True or false: 兹4 兹9 苷 6 5. If 兩z兩苷 1, describe the geometric relationship among the graphs of the fourth roots of z.

8. Does 兩a bi兩苷兹a2 (bi)2? 9. What is the conjugate of 3 5i?

冉

10. What is the modulus of 2 cos

冊

p p ? i sin 3 3

Chapter 5 Review Exercises In Exercises 1 to 4, write the complex number in standard form and give its conjugate.

1. 3 兹64

2. 兹4 6

3. 2 兹5

4. 5 兹27

In Exercises 5 to 20, simplify and write the complex number in standard form.

In Exercises 21 to 24, simplify and write each complex number as i, i, 1, or 1.

21. i 27

22. i 105

23.

i i 17

24. i 62

In Exercises 25 to 28, find the absolute value of each complex number.

25. 兩8i兩

26. 兩2 3i兩

27. 兩4 5i兩

28. 兩1 i兩

5. (兹4)(兹4)

6. (兹27)(兹3)

7. (3 7i) (2 5i)

8. (3 4i) (6 8i)

9. (6 8i) (9 11i)

10. 共3 5i兲共2 10i兲

In Exercises 29 to 32, write the complex number in trigonometric form.

12. 共2 3i兲共4 7i兲

29. z 苷 2 2i

30. z 苷兹3 i

31. z 苷 3 2i

32. z 苷 4 i

11. 共5 3i兲共2 5i兲 13.

2i 3 4i

15. i共2i兲共1 i兲2

14.

16. 共2 i兲3

17. 共3 兹4兲共3 兹16兲 18. 共2 兹9兲共3 兹81兲 19. 共2 兹3兲共2 兹3兲 20. 共3 兹5兲共2 兹5兲

4 i 7 2i

In Exercises 33 to 36, write the complex number in standard form. Round approximate constants to the nearest thousandth.

33. z 苷 5共cos 315° i sin 315°兲

冉

34. z 苷 6 cos

4p 4p i sin 3 3

冊

35. z 苷 2共cos 2 i sin 2兲 36. z 苷 3共cos 115° i sin 115°兲

Quantitative Reasoning

359

In Exercises 37 to 42, multiply the complex numbers. Write the answer in standard form.

In Exercises 49 to 54, find the indicated power. Write the answer in standard form.

37. 3共cos 225° i sin 225°兲 10共cos 45° i sin 45°兲

49. 关3共cos 45° i sin 45°兲兴 5

38. 5共cos 162° i sin 162°兲 2共cos 63° i sin 63°兲 39. 3共cos 12° i sin 12°兲 4共cos 126° i sin 126°兲 40. 共cos 23° i sin 23°兲 4共cos 233° i sin 233°兲 41. 3共cos 1.8 i sin 1.8兲 5共cos 2.5 i sin 2.5兲

50.

冋冉冊 cos

冉冊册

11p 11p i sin 8 8

8

51. 共1 i兹3兲7

52. 共2 2i兲10

53. 共兹2 i兹2兲5

54. 共3 4i兲5

42. 6共cos 3.1 i sin 3.1兲 5共cos 4.3 i sin 4.3兲

In Exercises 55 to 60, find the indicated roots. Write the answer in trigonometric form.

In Exercises 43 to 48, divide the complex numbers. Write the answer in trigonometric form.

55. The three cube roots of 27i

6(cos 50° i sin 50°) 43. 2(cos 150° i sin 150°) 45.

40(cos 66° i sin 66°) 8(cos 125° i sin 125°)

10(cos 3.7 i sin 3.7) 47. 6(cos 1.8 i sin 1.8)

30(cos 165° i sin 165°) 44. 10(cos 55° i sin 55°) 46.

2(cos 150° i sin 150°) 兹2(cos 200° i sin 200°)

4(cos 1.2 i sin 1.2) 48. 8(cos 5.2 i sin 5.2)

56. The four fourth roots of 8i 57. The four fourth roots of 256 58. The five fifth roots of 16兹2 16兹2i 59. The four fourth roots of 81 60. The three cube roots of 125

»» »»»» Quantitative Reasoning: Graphing the Mandelbrot Set PROGRAM: MANBROT Disp " THIS PROGRAM " Disp " GRAPHS THE TOP " Disp " HALF OF THE " Disp " MANDELBROT SET " Disp " " Disp " PRESS ENTER " Disp " TO START " Pause 24→I AxesOff:ClrDraw:FnOff –2→Xmin:1.03→Xmax 0→Ymin:2→Ymax For(A,–2,.4,.032) For(B,0,兹(4–A2),.032) Pt-On(A,B) (A+Bi)→S S→Z For (C,0,I) (Z2+S)→Z If abs(Z)>2 Then Pt-Off(A,B) I→C End:End:End:End StorePic 2

»»»

The T1-83/T1-83 Plus/T1-84 Plus program at the left graphs the top half of the Mandelbrot set. The bottom half of the Mandelbrot set is a reflection across the x-axis of the top half. See Figure 5.11 on page 357. The program uses the Mandelbrot iteration procedure, presented in the Exploring Concepts with Technology on page 356, to generate the iterates z1, z2, z3, … . In theory, a complex number z0 is defined to be an element of the Mandelbrot set if and only if the absolute value of each of its iterates is less than or equal to 2. In reality, a calculator or computer program cannot check an infinite number of iterates, but a fairly accurate graph can be produced by checking the first 24 iterates. If the program finds that the absolute values of the first 24 iterates of z0 are all less than or equal to 2, then the program assumes that z0 is an element of the Mandelbrot set and plots a point at z0 in the complex plane. If the program finds an iterate whose absolute value is greater than 2, then the program does not plot a point at z0. After the graph is produced, it is stored in memory, location Pic 2. You can recall the graph by pressing RecallPic 2. The RecallPic instruction is found in the DRAW STO menu.

360

Chapter 5

Complex Numbers QR1. Use the MANBROT program to graph the top half of the Mandelbrot set. The program requires about 25 minutes to complete the graph. The program step 24 l I sets the number of iterates to check. You can make the program run faster by using a natural number less than 24, but the graph will be less accurate. The graph will be slightly more accurate if you use a natural number larger than 24, but the program will run slower. QR2. Examine the graph from Exercise QR1 or Figure 5.11 to determine which of the following complex numbers are elements of the Mandelbrot set. The WINDOW settings for the program are Xmin = –2, Xmax = 1.03, Ymin = 0, and Ymax = 2. 0.25 0.25i, 1 0.1i, 0.75 0.75i, 0.1 0.2i QR3. Examine the iterates of 2 to verify that 2 is an element of the Mandelbrot set. QR4. What is the first iterate of 2i? Is 2i an element of the Mandelbrot set? QR5. Which step in the program applies the Mandelbrot iteration procedure? QR6. Search the Internet for additional information on the Mandelbrot set. Describe two properties of the Mandelbrot set that you find the most interesting.

Chapter 5 Test 1. Write 6 兹9 in the form a bi. 2. Simplify: 兹18 3. Simplify 共3 兹4兲共7 兹9兲. Write the answer in standard form. 4. Simplify 共1 兹25兲共8 兹16兲. Write the answer in standard form. 5. Simplify: 共兹12兲共兹3兲 6. Simplify: i 263 7. Simplify: 共3 7i兲共2 9i兲 8. Simplify: 共6 9i兲共4 3i兲 9. Simplify: 共3 5i兲共3 5i兲 10. Simplify:

4 5i i

11. Simplify:

2 7i 4 3i

12. Simplify:

6 2i 1i

13. Find the absolute value of 3 5i. 14. Write 3 3i in trigonometric form. 15. Write 6i in trigonometric form. 16. Write 4共cos 120° i sin 120°兲 in standard form. 17. Write 5共cos 225° i sin 225°兲 in standard form. 18. Simplify 3共cos 28° i sin 28°兲 4共cos 17° i sin 17°兲. Write the answer in standard form. 19. Simplify 5共cos 115° i sin 115°兲 4共cos 10° i sin 10°兲. Write the answer in standard form.

361

Cumulative Review Exercises 24(cos 258° i sin 258°) . Write the answer in 6(cos 78° i sin 78°) standard form.

In Exercises 23 to 25, write the indicated roots and solutions in trigonometric form.

18(cos 50° i sin 50°) . Write the answer in 3(cos 140° i sin 140°) standard form.

24. Find the three cube roots of 1 i兹3.

20. Simplify

21. Simplify

23. Find the six sixth roots of 64.

25. Find the five solutions of z 5 32 苷 0.

22. Simplify 共2 2i兹3兲12. Write the answer in standard form.

Cumulative Review Exercises 1. Solve x 2 x 6 0. Write the answer using interval notation. 2. What is the domain of f共x兲苷

x2 ? x2 4

3. Find c in the domain of f共x兲苷

4 12 in Quadrant I and cos b 苷 in Quad5 13 rant IV, find cos共a b兲.

13. Given sin a 苷

x such that f共c兲苷 2. x1

x2 1 4. Given f共x兲苷 sin 3x and g共x兲苷 , find 共f ⴰ g兲共x兲. 3 x 5. Given f共x兲苷 , find f 1共3兲. x1 6. Convert

3p radians to degrees. 2

7. Find the length of the hypotenuse a for the right triangle shown at the right.

12. Express sin 2x cos 3x sin 3x cos 2x in terms of the sine function.

冋冉冊

14. Find the exact value of sin sin1

冉冊册

3 5 cos1 5 13

.

15. Solve sin 2x 苷兹3 sin x for 0 x 2p. 16. For the triangle at the right, find the length of side c. Round to the nearest ten.

78° 140 cm

130 cm

a c

38° 20 cm

8. If t is any real number, what are the values of a and b for the inequality a sin t b? 9. Graph y 苷 3 sin px. 10. Graph y 苷

px 1 tan . 2 4

11. Verify the identity

sin x 苷 csc x cot x. 1 cos x

17. Find the angle between the vectors v 苷 3i 2j and w 苷 5i 3j. Round to the nearest tenth of a degree. 18. WORK A force of 100 pounds on a rope is used to drag a box up a ramp that is inclined 15°. If the rope makes an angle of 30° with the ground, find the work done in moving the box 15 feet along the ramp. Round to the nearest foot-pound. 19. Write the complex number 2 2i in trigonometric form. 20. Find the three cube roots of 27. Write the roots in standard form.

6

Topics in Analytic Geometry

Circle

Ellipse

6.1

Parabolas

6.2

Ellipses

6.3

Hyperbolas

6.4

Rotation of Axes

6.5

Introduction to Polar Coordinates

6.6

Polar Equations of the Conics

6.7

Parametric Equations

Conic Sections and Their Applications

Parabola

The study of the geometric figures called conic sections is one of the topics of this chapter. Each of these figures is formed by the intersection of a plane and a cone.1 Hyperbola The ancient Greeks were the first to study the conic sections. Their study was motivated by the many new and interesting mathematical concepts they were able to discover, without regard to finding or producing practical applications. The ancient Greeks would be surprised to learn that their study of the conic sections helped produce a body of knowledge with many practical applications in several diverse fields, including astronomy, architecture, engineering, and satellite communications. Exercise 55 on page 386 illustrates a medical application of conic sections, and Exercise 47 on page 373 illustrates an application of conic sections in the design of a ski with a parabolic sidecut.

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

362

1 Only

one branch of the hyperbola in the figure is displayed; however, all hyperbolas have two branches that are formed by the intersection of a plane and a double-napped cone as shown in Figure 6.1 on page 363.

6.1

■

■

363

Parabolas

Section 6.1 ■

Parabolas

The graph of a circle, an ellipse, a parabola, or a hyperbola can be formed by the intersection of a plane and a cone. Hence these figures are referred to as conic sections. See Figure 6.1.

Parabolas with Vertex at (0, 0) Parabolas with Vertex at (h, k) Applications

Axis

Axis

Axis

Hyperbola

E Ellipse Circle C

Parabola

take note

Figure 6.1

If the intersection of a plane and a cone is a point, a line, or two intersecting lines, then the intersection is called a degenerate

conic section.

Cones intersected by planes

A plane perpendicular to the axis of the cone intersects the cone in a circle (plane C). The plane E, tilted so that it is not perpendicular to the axis, intersects the cone in an ellipse. When the plane is parallel to a line on the surface of the cone, the plane intersects the cone in a parabola. When the plane intersects both portions of the cone, a hyperbola is formed.

Parabolas with Vertex at (0, 0)

Math Matters Appollonius (262–200 B.C.) wrote an eight-volume treatise entitled On Conic Sections in which he derived the formulas for all the conic sections. He was the first to use the words parabola, ellipse, and hyperbola.

In addition to the geometric description of a conic section just given, a conic section can be defined as a set of points. This method uses specified conditions about a curve to determine which points in the coordinate system are points of the graph. For example, a parabola can be defined by the following set of points. Definition of a Parabola A parabola is the set of points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus) not on the directrix.

Axis of symmetry

y

Focus x Directrix Vertex

Figure 6.2

The line that passes through the focus and is perpendicular to the directrix is called the axis of symmetry of the parabola. The midpoint of the line segment between the focus and directrix on the axis of symmetry is the vertex of the parabola, as shown in Figure 6.2. Using this definition of a parabola, we can determine an equation of a parabola. Suppose that the coordinates of the vertex of a parabola are V共0, 0兲 and the axis of symmetry is the y-axis. The equation of the directrix is y 苷 p, p 0. The focus lies on the axis of symmetry and is the same distance from the vertex as the vertex is from the directrix. Thus the coordinates of the focus are F共0, p兲, as shown in Figure 6.3.

364

Chapter 6

Topics in Analytic Geometry

y

Focus F(0, p) P(x, y)

x

V(0, 0)

Directrix y = −p

Let P共x, y兲 be any point P on the parabola. Then, using the distance formula and the fact that the distance between any point P on the parabola and the focus is equal to the distance from the point P to the directrix, we can write the equation d共P, F兲苷 d共P, D兲 By the distance formula, 兹共x 0兲2 共 y p兲2 苷 y p Now, squaring each side and simplifying, we get

共兹共x 0兲2 共 y p兲2 兲2 苷共 y p兲2

D(x, −p)

x 2 y 2 2py p2 苷 y 2 2py p2 x 2 苷 4py This is the standard form of the equation of a parabola with vertex at the origin and the y-axis as its axis of symmetry. The standard form of the equation of a parabola with vertex at the origin and the x-axis as its axis of symmetry is derived in a similar manner.

Figure 6.3

TO REVIEW

Axis of Symmetry See page 54.

Standard Forms of the Equation of a Parabola with Vertex at the Origin

Axis of Symmetry Is the y-Axis The standard form of the equation of a parabola with vertex 共0, 0兲 and the y-axis as its axis of symmetry is x 2 苷 4py The focus is 共0, p兲, and the equation of the directrix is y 苷 p. If p 0, the graph of the parabola opens up. See Figure 6.4a. If p 0, the graph of the parabola opens down. See Figure 6.4b.

take note The tests for y-axis and x-axis symmetry can be used to verify

Axis of Symmetry Is the x-Axis

these statements and provide

The standard form of the equation of a parabola with vertex 共0, 0兲 and the x-axis as its axis of symmetry is y 2 苷 4px The focus is 共p, 0兲, and the equation of the directrix is x 苷 p. If p 0, the graph of the parabola opens to the right. See Figure 6.4c. If p 0, the graph of the parabola opens to the left. See Figure 6.4d.

connections to earlier topics on symmetry.

y x2 = 4py, p > 0

d(P, F) = d(P, D) Focus F(0, p)

p

p Directix y = −p

|p|

a. The graph of x 2 苷 4py with p 0

P(x, y)

P(x, y) d(P, F) = d(P, D)

|p|

x

D(x, −p)

D(−p, y)

D(x, −p)

Directix y = −p

P(x, y) Vertex V(0, 0)

y

y

y

x2 = 4py, p < 0

x P(x, y)

Focus F(0, p) d(P, F) = d(P, D)

b. The graph of x 2 苷 4py with p 0

p

Directix x = −p

x

p

y2 = 4px, p > 0

c. The graph of y 2 苷 4px with p 0

Figure 6.4

d(P, F) = d(P, D) Focus F(p, 0)

Focus F(p, 0)

Vertex V(0, 0)

Vertex V(0, 0)

D(−p, y)

Vertex V(0, 0) |p|

y2 = 4px, p < 0

x

|p|

Directix x = −p

d. The graph of y 2 苷 4px with p 0

6.1 QUESTION

365

Parabolas

Does the graph of y 2 苷 4x open up, down, to the left, or to the right?

EXAMPLE 1

»

Find the Focus and Directrix of a Parabola

Find the focus and directrix of the parabola given by the equation 1 y 苷 x 2. 2

Solution Because the x term is squared, the standard form of the equation is x 2 苷 4py. 1 2 x 2 x 2 苷 2y y苷

• Write the given equation in standard form.

Comparing this equation with x 2 苷 4py gives 4p 苷 2 1 p苷 2 Because p is negative, the parabola opens down, and the focus is below the vertex 共0, 0兲, as shown in Figure 6.5. The coordinates of the 1 focus are 0, . The equation of the 2 1 directrix is y 苷 . 2

冉冊

y

y=

–4

(

1 2

4 0, − 1 2

x

)

–8

Figure 6.5

»

Try Exercise 6, page 371

EXAMPLE 2

»

Find the Equation of a Parabola in Standard Form

Find the equation in standard form of the parabola with vertex at the origin and focus at 共2, 0兲.

Solution Because the vertex is 共0, 0兲 and the focus is at 共2, 0兲, p 苷 2. The graph of the parabola opens toward the focus, so in this case the parabola opens to Continued ANSWER

The graph opens to the left.

䉴

366

Chapter 6

Topics in Analytic Geometry the left. The equation in standard form of the parabola that opens to the left is y 2 苷 4px. Substitute 2 for p in this equation and simplify. y 2 苷 4共2兲x 苷 8x The equation of the parabola is y 2 苷 8x.

»

Try Exercise 30, page 371

Integrating Technology 9

−9

2

The graph of y 2 苷 8x is shown in Figure 6.6. Note that the graph is not the graph of a function. To graph y 2 苷 8x with a graphing utility, we first solve for y to produce y 苷兹8x. From this equation we can see that for any x 0, there are two values of y. For example, when x 苷 2, y 苷兹共8兲共2兲苷兹16 苷 4

−9

The graph of y 2 苷 8x in Figure 6.6 was constructed by graphing both

Figure 6.6

y

Y1 苷兹8x and Y2 苷兹8x in the same window.

y' h

(x, y) (x', y')

x'

y'

x'

(h, k) k

x

(0, 0)

Figure 6.7

Parabolas with Vertex at (h, k) The equation of a parabola with a vertical or horizontal axis of symmetry and with vertex at a point 共h, k兲 can be found by using the translations discussed previously. Consider a coordinate system with coordinate axes labeled x and y placed so that its origin is at 共h, k兲 of the xy-coordinate system. The relationship between an ordered pair in the x y -coordinate system and in the xy-coordinate system is given by the transformation equations x 苷 x h y 苷 y k

(1)

Now consider a parabola with vertex at 共h, k兲, as shown in Figure 6.7. Create a new coordinate system with axes labeled x and y and with its origin at 共h, k兲. The equation of a parabola in the x y -coordinate system is 共x 兲2 苷 4py

(2)

Using the transformation Equations (1), we can substitute the expressions for x and y into Equation (2). The standard form of the equation of a parabola with vertex 共h, k兲 and a vertical axis of symmetry is 共x h兲2 苷 4p共 y k兲 Similarly, we can derive the standard form of the equation of a parabola with vertex 共h, k兲 and a horizontal axis of symmetry.

6.1

367

Parabolas

Standard Forms of the Equation of a Parabola with Vertex at (h, k)

Vertical Axis of Symmetry y

y'

The standard form of the equation of a parabola with vertex 共h, k兲 and a vertical axis of symmetry is F(h, k + p)

V(h, k)

共x h兲2 苷 4p共 y k兲 x'

The focus is 共h, k p兲, and the equation of the directrix is y 苷 k p. If p 0, the parabola opens up. See Figure 6.8. If p 0, the parabola opens down.

Horizontal Axis of Symmetry

Directrix y=k−p x

The standard form of the equation of a parabola with vertex 共h, k兲 and a horizontal axis of symmetry is

Figure 6.8

共 y k兲2 苷 4p共x h兲 The focus is 共h p, k兲, and the equation of the directrix is x 苷 h p. If p 0, the parabola opens to the right. If p 0, the parabola opens to the left. In Example 3 we complete the square to find the standard form of a parabola, and then use the standard form to determine the vertex, focus, and directrix of the parabola. EXAMPLE 3

»

Find the Focus and Directrix of a Parabola

Find the equation of the directrix and the coordinates of the vertex and focus of the parabola given by the equation 3x 2y 2 8y 4 苷 0.

Solution Rewrite the equation so that the y terms are on one side of the equation, and then complete the square on y. TO REVIEW

Completing the Square See page 26.

3x 2y 2 8y 4 苷 0 2y 2 8y 苷 3x 4 2共 y 2 4y兲苷 3x 4 2 2共 y 4y 4兲苷 3x 4 8 2共 y 2兲2 苷 3共x 4兲 3 共y 2兲2 苷共x 4兲 2

• Complete the square. Note that 2 4 8 is added to each side. • Simplify and factor. • Write the equation in standard form.

Comparing this equation to 共 y k兲2 苷 4p共x h兲, we have a parabola that 3 3 opens to the left with vertex 共4, 2兲 and 4p 苷 . Thus p 苷 . 2 8 The coordinates of the focus are

冉冉冊冊冉冊 4

3 29 , 2 苷 , 2 8 8

Continued

䉴

368

Chapter 6

Topics in Analytic Geometry The equation of the directrix is

y (−2, 1) 2

冉冊

x苷4 6 x

−6

(

29 , −2 8

)

(4, −2)

(– 2, – 5) x=

−6

35 8

3 8

苷

35 8

Choosing some values for y and finding the corresponding values for x, we plot a few points. Because the line y 苷 2 is the axis of symmetry, for each point on one side of the axis of symmetry there is a corresponding point on the other side. Two points are 共2, 1兲 and 共2, 5兲. See Figure 6.9.

»

Try Exercise 22, page 371

Figure 6.9

EXAMPLE 4

»

Find the Equation in Standard Form of a Parabola

Find the equation in standard form of the parabola with directrix x 苷 1 and focus 共3, 2兲.

Solution

y (1, 2)

(−1, 2)

The vertex is the midpoint of the line segment joining the focus 共3, 2兲 and the point 共1, 2兲 on the directrix. (3, 2)

2

5 x = −1

Figure 6.10

共h, k兲苷

x

冉

冊

1 3 2 2 , 苷共1, 2兲 2 2

The standard form of the equation is 共 y k兲2 苷 4p共x h兲. The distance from the vertex to the focus is 2. Thus 4p 苷 4共2兲苷 8, and the equation of the parabola in standard form is 共 y 2兲2 苷 8共x 1兲. See Figure 6.10.

»

Try Exercise 32, page 371

Applications A principle of physics states that when light is reflected from a point P on a surface, the angle of incidence (that of the incoming ray) equals the angle of reflection (that of the outgoing ray). See Figure 6.11. This principle applied to parabolas has some useful consequences.

Reflected beam

Incident beam

Angle of reflection

Angle of incidence P

Figure 6.11

6.1

369

Parabolas

Reflective Property of a Parabola The line tangent to a parabola at a point P makes equal angles with the line through P and parallel to the axis of symmetry and the line through P and the focus of the parabola (see Figure 6.12).

α

F

β

P

A cross section of the reflecting mirror of a telescope has the shape of a parabola. The incoming parallel rays of light are reflected from the surface of the mirror to the eyepiece. See Figure 6.13. Flashlights and car headlights also make use of this reflective property. The light bulb is positioned at the focus of the parabolic reflector, which causes the reflected light to be reflected outward in parallel rays. See Figure 6.14.

Tangent at P

a苷b

Figure 6.12

Eyepiece

Light at focus

Parabolic mirror

Parallel rays

Figure 6.13

Figure 6.14

When a parabola is revolved about its axis, it produces a three-dimensional surface called a paraboloid. The focus of a paraboloid is the same as the focus of the parabola that was revolved to generate the paraboloid. The vertex of a paraboloid is the same as the vertex of the parabola that was revolved to generate the paraboloid. In Example 5 we find the focus of a satellite dish that has the shape of a paraboloid.

EXAMPLE 5 Incoming signal

Axis (parallel to incoming signal)

Parabolic satellite dish

Receiver located at the focus

Figure 6.15

»

Find the Focus of a Satellite Dish

A satellite dish has the shape of a paraboloid. The signals that it receives are reflected to a receiver that is located at the focus of the paraboloid. If the dish is 8 feet across at its opening and 1.25 feet deep at its center, determine the location of its focus.

Solution Figure 6.15 shows that a cross section of the paraboloid along its axis of symmetry is a parabola. Figure 6.16 shows this cross section placed in a

rectangular coordinate system with the vertex of the parabola at 共0, 0兲 and the axis of symmetry of the parabola on the y-axis. The parabola has an equation of the form 4py 苷 x 2 Continued

䉴

370

Chapter 6

Topics in Analytic Geometry Because the parabola contains the point 共4, 1.25兲, this equation is satisfied by the substitutions x 苷 4 and y 苷 1.25. Thus we have 4p共1.25兲苷 42 5p 苷 16 16 p苷 5

y Focus (0, p)

Satellite dish

2

(– 4, 1.25)

–4

4

(4, 1.25) –2

2

4

x

Figure 6.16

The focus of the satellite dish is on the axis of symmetry of the dish, and it is 1 3 feet above the vertex of the dish. See Figure 6.16. 5

»

Try Exercise 40, page 372

Topics for Discussion 1. Do the graphs of the parabola given by y 苷 x 2 and the vertical line given by x 苷 10,000 intersect? Explain. 2. A student claims that the focus of the parabola given by y 苷 8x 2 is at 共0, 2兲 because 4p 苷 8 implies that p 苷 2. Explain the error in the student’s reasoning. 3. The vertex of a parabola is always halfway between its focus and its directrix. Do you agree? Explain. 4. A tutor claims that the graph of 共x h兲2 苷 4p共 y k兲 has a y-intercept of h2 0, k . Explain why the tutor is correct. 4p

冉

冊

Exercise Set 6.1 1. Examine the following four equations and the graphs labeled i, ii, iii, and iv. Determine which graph is the graph of each equation.

i.

28

−63.3 −9.1

a. y 2 苷 x

−28 Xscl=10, Yscl=10

−2.5

iii.

21.6

9.1

b. x 2 苷 4y 1 c. x 2 苷 y 2

ii.

9.5

iv.

6

2.5 −9.1

d. y 2 苷 12x

−5.6

9.1

12.6

−6

−9.5

6.1 2. Examine the following four equations and the graphs labeled i, ii, iii, and iv. Determine which graph is the graph of each equation. a. 共y 3兲2 苷 x 2 c.

b. 共y 3兲2 苷 2共x 2兲

1 共y 3兲苷共x 2兲2 2

i.

V(2, 3)

− 9.4

6.2

32. Find the equation in standard form of the parabola with vertex at 共2, 3兲 and focus 共0, 3兲.

−9.4

9.4

iv.

6.2

6.2 V(2, 3)

V(2, 3) −9.4

9.4

−9.4

9.4

−6.2

−6.2

In Exercises 3 to 28, find the vertex, focus, and directrix of the parabola given by each equation. Sketch the graph.

13. x 2 苷 4y

4. 2y 2 苷 x

1 x 3

6. x 2 苷

5. y 2 苷

7. 共x 2兲2 苷 8共 y 3兲 9. 共 y 4兲2 苷 4共x 2兲

33. Find the equation in standard form of the parabola with focus 共3, 3兲 and directrix y 苷 5. 34. Find the equation in standard form of the parabola with focus 共2, 4兲 and directrix x 苷 4.

−6.2

− 6.2

iii.

30. Find the equation in standard form of the parabola with vertex at the origin and focus 共5, 0兲. 31. Find the equation in standard form of the parabola with vertex at 共1, 2兲 and focus 共1, 3兲.

V(2, 3) 9.4

29. Find the equation in standard form of the parabola with vertex at the origin and focus 共0, 4兲.

d. 2共y 3兲苷共x 2兲2

ii.

6.2

371

Parabolas

1 y 4

8. 共 y 1兲2 苷 6共x 1兲 10. 共x 3兲2 苷共 y 2兲

11. 共 y 1兲2 苷 2x 8

12. 共x 2兲2 苷 3y 6

13. 共2x 4兲2 苷 8y 16

14. 共3x 6兲2 苷 18y 36

15. x 2 8x y 6 苷 0

16. x 2 6x y 10 苷 0

17. x y 2 3y 4 苷 0

18. x y 2 4y 9 苷 0

19. 2x y 2 6y 1 苷 0

20. 3x y 2 8y 4 苷 0

21. x 2 3x 3y 1 苷 0

22. x 2 5x 4y 1 苷 0

23. 2x 2 8x 4y 3 苷 0

24. 6x 3y 2 12y 4 苷 0

25. 2x 4y 2 8y 5 苷 0

26. 4x 2 12x 12y 7 苷 0

27. 3x 2 6x 9y 4 苷 0

28. 2x 3y 2 9y 5 苷 0

35. Find the equation in standard form of the parabola that has vertex 共4, 1兲, has its axis of symmetry parallel to the y-axis, and passes through the point 共2, 2兲. 36. Find the equation in standard form of the parabola that has vertex 共3, 5兲, has its axis of symmetry parallel to the x-axis, and passes through the point 共4, 3兲. 37. STRUCTURAL DEFECTS Ultrasound is used as a nondestructive method of determining whether a support beam for a structure has an internal fracture. In one scanning procedure, if the resulting image is a parabola, engineers know that there is a structural defect. Suppose that a scan produced an image whose equation is x 苷 0.325y 2 13y 120 Determine the vertex and focus of the graph of this parabola. 38. FOUNTAIN DESIGN A fountain in a shopping mall has two parabolic arcs of water intersecting as shown below. The equation of one parabola is y 苷 0.25x 2 2x and the equation of the second parabola is y 苷 0.25x 2 4.5x 16.25 How high above the base of the fountain do the parabolas intersect? All dimensions are in feet. y

x

372

Chapter 6

Topics in Analytic Geometry a. Find an equation of a cross section of the paraboloid that passes through the vertex of the paraboloid. Assume that the dish has its vertex at 共0, 0兲 and a vertical axis of symmetry.

39. SATELLITE DISH A satellite dish has the shape of a paraboloid. The signals that it receives are reflected to a receiver that is located at the focus of the paraboloid. If the dish is 8 feet across at its opening and 1 foot deep at its vertex, determine the location (distance above the vertex of the dish) of its focus. 40. RADIO TELESCOPES The antenna of a radio telescope is a paraboloid measuring 81 feet across with a depth of 16 feet. Determine, to the nearest tenth of a foot, the distance from the vertex to the focus of this antenna.

b. Find the depth of the dish. Round to the nearest foot. 43.

y Parabolic cross section

The surface area of a paraboloid with radius r and pr depth d is given by S 苷 2 关共r 2 4d 2 兲3/2 r 3 兴. 6d Approximate (to the nearest 100 square feet) the surface area of

16 ft x 81 ft

41. CAPTURING SOUND During televised football games, a parabolic microphone is used to capture sounds. The shield of the microphone is a paraboloid with a diameter of 18.75 inches and a depth of 3.66 inches. To pick up the sounds, a microphone is placed at the focus of the paraboloid. How far (to the nearest tenth of an inch) from the vertex of the paraboloid should the microphone be placed?

a. the radio telescope in Exercise 40. b. the Lovell Telescope (see Exercise 42). 44. THE HALE TELESCOPE The parabolic mirror in the Hale telescope at the Palomar Observatory in southern California has a diameter of 200 inches and a concave depth of 3.75375 inches. Determine the location of its focus (to the nearest inch). y Focus (0, p) Not drawn to scale Parabolic mirror

(–100, 3.75375)

(100, 3.75375)

3.75375 –100

–50

50

100

x

200 in.

Cross Section of the Mirror in the Hale Telescope

42.

THE LOVELL TELESCOPE The Lovell Telescope is a radio telescope located at the Jodrell Bank Observatory in Cheshire, England. The dish of the telescope has the shape of a paraboloid with a diameter of 250 feet. The distance from the vertex of the dish to its focus is 75 feet.

45. THE LICK TELESCOPE The parabolic mirror in the Lick telescope at the Lick Observatory on Mount Hamilton has a diameter of 120 inches and a focal length of 600 inches. (Note: The focal length of a parabolic mirror is the distance from the vertex of the mirror to the mirror’s focus.) In the construction of the mirror, workers ground the mirror as shown in the following diagram. Determine the dimension a, which is the concave depth of the mirror. y Focus (0, 600) Not drawn to scale Parabolic mirror

600 in. (–60, a)

(60, a)

a –60

–30

30

60

x

120 in.

Cross Section of the Mirror in the Lick Telescope

6.1 46. HEADLIGHT DESIGN A light source is to be placed on the axis of symmetry of the parabolic reflector shown in the figure below. How far to the right of the vertex point should the light source be located if the designer wishes the reflected light rays to form a beam of parallel rays?

373

Parabolas y

(−800, 53)

parabolic sidecut V(0, 32) x

tail Vertex point

waist

shovel

a. Find the equation in standard form of the parabolic sidecut.

6 in. Axis of symmetry

Parabolic reflector

b. How wide is the ski at its shovel (the widest point near the front of the ski), where x 苷 900? Round to the nearest millimeter.

6 in.

47. SKI DESIGN Many contemporary skis have parabolic sidecuts that allow a skier to carve tighter turns than are possible with traditional skis. In the following diagram, x is the directed distance to the right or left of the y-axis (with x and y measured in millimeters). The x-axis is on the center horizontal axis of the ski. The vertex of the parabolic sidecut is V共0, 32兲.

48.

The only information we have about a particular parabola is that 共2, 3兲 and 共2, 3兲 are points on the parabola. Explain why it is not possible to find the equation of this particular parabola using just this information.

Connecting Concepts In Exercises 49 to 51, use the following definition of latus rectum: The line segment that has endpoints on a parabola, passes through the focus of the parabola, and is perpendicular to the axis of symmetry is called the latus rectum of the parabola. y

The result of Exercise 51 can be stated as the following theorem: Two points on a parabola will be 2円p円 units on each side of the axis of symmetry on the line through the focus and perpendicular to that axis.

52. Use the theorem to sketch a graph of the parabola given by the equation 共x 3兲2 苷 2共 y 1兲. 53. Use the theorem to sketch a graph of the parabola given by the equation 共 y 4兲2 苷共x 1兲.

Latus rectum

54. By using the definition of a parabola, find the equation in standard form of the parabola with V共0, 0兲, F共c, 0兲, and directrix x 苷 c.

F(0, p)

x

49. Find the length of the latus rectum for the parabola x 2 苷 4y. 50. Find the length of the latus rectum for the parabola y 2 苷 8x. 51. Find the length of the latus rectum for any parabola in terms of 兩 p兩, the distance from the vertex of the parabola to its focus.

55. Sketch a graph of 4共 y 2兲苷 x兩x兩 1. 56. Find the equation of the directrix of the parabola with vertex at the origin and focus at the point 共1, 1兲. 57. Find the equation of the parabola with vertex at the origin and focus at the point 共1, 1兲. (Hint: You will need the answer to Exercise 56 and the definition of a parabola.)

374

Chapter 6

Topics in Analytic Geometry

Projects 1.

3-D OPTICAL ILLUSION In the photo below, a strawberry appears to float above a black disc.

If you try to touch the strawberry, you will discover that the strawberry is not located where it appears. Search the Internet to discover how parabolic mirrors are used to create this illusion. Write a few sentences, along with a diagram that explains how the illusion is accomplished.

Mirage by InnovaToys Photo source: http://innovatoys.com/ c/OPT

Section 6.2 ■

■

■ ■

Ellipses with Center at (0, 0) Ellipses with Center at (h, k) Eccentricity of an Ellipse Applications

Ellipses P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A25.

PS1. Find the midpoint and the length of the line segment between P1共5, 1兲 and P2共1, 5兲. [1.2] PS2. Solve: x 2 6x 16 苷 0 [1.1] PS3. Solve: x 2 2x 苷 2 [1.1] PS4. Complete the square of x 2 8x and write the result as the square of a binomial. [1.2] PS5. Solve 共x 2兲2 y 2 苷 4 for y. [1.2] PS6. Graph: 共x 2兲2 共y 3兲2 苷 16 [1.2]

An ellipse is another of the conic sections formed when a plane intersects a right circular cone. If b is the angle at which the plane intersects the axis of the cone and

6.2

take note If a plane intersects a cone at the

Ellipses

375

a is the angle shown in Figure 6.17, an ellipse is formed when a b 90 . If b 苷 90 , then a circle is formed.

vertex of the cone so that the resulting figure is a point, the point

Axis

Axis

β

is called a degenerate ellipse. See the accompanying figure.

β l1

l1

α

Ellipse

α Vertex

Circle

l2

l2

Figure 6.17 Degenerate ellipse

As was the case for a parabola, there is a definition for an ellipse in terms of a certain set of points in the plane.

Definition of an Ellipse An ellipse is the set of all points in the plane the sum of whose distances from two fixed points (foci) is a positive constant.

Equipped only with a piece of string and two tacks, we can use this definition to draw an ellipse (see Figure 6.18). Tack the ends of the string to the foci, and trace a curve with a pencil held tight against the string. The resulting curve is an ellipse. The positive constant mentioned in the definition of an ellipse is the length of the string. F1

F2

Figure 6.18

Ellipses with Center at (0, 0) The graph of an ellipse has two axes of symmetry (see Figure 6.19). The longer axis is called the major axis. The foci of the ellipse are on the major axis. The shorter axis is called the minor axis. It is customary to denote the length of the major axis by 2a and the length of the minor axis by 2b. The center of the ellipse is the midpoint of the major axis. The endpoints of the major axis are the vertices (plural of vertex) of the ellipse. A semimajor axis of an ellipse is a line segment that connects the center point of the ellipse with a vertex. Its length is half the length of the major axis. A semiminor axis of an ellipse is a line segment that lies on the minor axis and connects the center point with a point on the ellipse. Its length is half the length of the minor axis.

376

Chapter 6

Topics in Analytic Geometry y y

Minor axis

Vertex

a−c Vertex

Focus

Major axis

F1(−c, 0)

Focus

x a+c

x

Center

V2 (a, 0)

Figure 6.19

P(x, y)

F2(c, 0)

Figure 6.20

Consider the point V2共a, 0兲, which is one vertex of an ellipse, and the points F2共c, 0兲 and F1共c, 0兲, which are the foci of the ellipse shown in Figure 6.20. The distance from V2 to F1 is a c. Similarly, the distance from V2 to F2 is a c. From the definition of an ellipse, the sum of the distances from any point on the ellipse to the foci is a positive constant. By adding the expressions a c and a c, we have

y

共a c兲共a c兲苷 2a F1(− c, 0)

F2(c, 0)

x

Thus the positive constant referred to in the definition of an ellipse is 2a, the length of the major axis. Now let P共x, y兲 be any point on the ellipse (see Figure 6.21). By using the definition of an ellipse, we have d共P, F1 兲 d共P, F2 兲苷 2a

Figure 6.21

兹共x c兲2 y 2 兹共x c兲2 y 2 苷 2a Subtract the second radical from each side of the equation, and then square each side.

关兹共x c兲2 y 2 兴2 苷关 2a 兹共x c兲2 y 2 兴2 共x c兲2 y 2 苷 4a2 4a兹共x c兲2 y 2 共x c兲2 y 2 x 2 2cx c 2 y 2 苷 4a2 4a兹共x c兲2 y 2 x 2 2cx c 2 y 2 4cx 4a2 苷 4a兹共x c兲2 y 2 2 关cx a2 兴 2 苷关 a兹共x c兲2 y 2 兴

• Divide by 4, and then square each side.

c2x 2 2cxa2 a4 苷 a2x 2 2cxa2 a2c 2 a2y 2 a2x 2 c2x 2 a2y 2 苷 a4 a2c 2

• Rewrite with x and y terms on the left side.

共a2 c2 兲x 2 a2y 2 苷 a2共a2 c 2 兲 b2x 2 a2y 2 苷 a2b2 x2 y2 苷1 a2 b2

• Factor. • Let b 2 a 2 c 2. • Divide each side by a 2b 2. The result is an equation of an ellipse with center at (0, 0).

6.2 y

Standard Forms of the Equation of an Ellipse with Center at the Origin

(0, b) (a, 0) (− a, 0)

(− c, 0)

377

Ellipses

(c, 0)

x

Major Axis on the x-Axis The standard form of the equation of an ellipse with center at the origin and major axis on the x-axis (see Figure 6.22a) is given by

(0, − b)

x2 y2 苷 1, 2 a b2

a. Major axis on x-axis y (0, a)

ab

The length of the major axis is 2a. The length of the minor axis is 2b. The coordinates of the vertices are 共a, 0兲 and 共a, 0兲, and the coordinates of the foci are 共c, 0兲 and 共c, 0兲, where c 2 苷 a2 b2.

(0, c)

Major Axis on the y-Axis (− b, 0)

(b, 0)

x

The standard form of the equation of an ellipse with center at the origin and major axis on the y-axis (see Figure 6.22b) is given by

(0, − c)

x2 y2 苷 1, b2 a2

(0, − a)

b. Major axis on y-axis

Figure 6.22

ab

The length of the major axis is 2a. The length of the minor axis is 2b. The coordinates of the vertices are 共0, a兲 and 共0, a兲, and the coordinates of the foci are 共0, c兲 and 共0, c兲, where c 2 苷 a2 b2.

QUESTION

For the graph of

x2 y2 苷 1, is the major axis on the x-axis or 16 25

the y-axis?

EXAMPLE 1

»

Find the Vertices and Foci of an Ellipse

Find the vertices and foci of the ellipse given by the equation

x2 y2 苷 1. 25 49

Sketch the graph.

Solution Because the y 2 term has the larger denominator, the major axis is on the y-axis. a2 苷 49 a苷7

b2 苷 25 b苷5

c 2 苷 a 2 b2 苷 49 25 苷 24 c 苷兹24 苷 2兹6 Continued

ANSWER

Because 25 16, the major axis is on the y-axis.

䉴

378

Chapter 6 y

Topics in Analytic Geometry The vertices are 共0, 7兲 and 共0, 7兲. The foci are 共 0, 2兹6 兲 and 共 0, 2兹6 兲. See

(0, 7)

Figure 6.23.

»

( 0, 2 6)

Try Exercise 22, page 385

6 x

−6

An ellipse with foci 共3, 0兲 and 共3, 0兲 and major axis of length 10 is shown in Figure 6.24. To find the equation of the ellipse in standard form, we must find a2 and b2. Because the foci are on the major axis, the major axis is on the x-axis. The length of the major axis is 2a. Thus 2a 苷 10. Solving for a, we have a 苷 5 and a2 苷 25. Because the foci are 共3, 0兲 and 共3, 0兲 and the center of the ellipse is the midpoint between the two foci, the distance from the center of the ellipse to a focus is 3. Therefore, c 苷 3. To find b2, use the equation

(0, −2 6) (0, − 7)

x2 y2 苷1 25 49 Figure 6.23

y 5

(5, 0) (−5, 0)

(3, 0)

(−3, 0)

x

−5

x2 y2 苷1 25 16

c 2 苷 a 2 b2 9 苷 25 b2 b2 苷 16

Figure 6.24

The equation of the ellipse in standard form is

x2 y2 苷 1. 25 16

Ellipses with Center at (h, k) y

y'

The equation of an ellipse with center at 共h, k兲 and with a horizontal or vertical major axis can be found by using a translation of coordinates. On a coordinate system with axes labeled x and y , the standard form of the equation of an ellipse with center at the origin of the x y -coordinate system is

x' = x − h y' = y − k (x', y')

C(h, k)

x'

x

(0, 0)

Figure 6.25

共x 兲2 共 y 兲2 2 苷1 a2 b Now place the origin of the x y -coordinate system at 共h, k兲 in an xy-coordinate system. See Figure 6.25. The relationship between an ordered pair in the x y -coordinate system and one in the xy-coordinate system is given by the transformation equations x 苷 x h y 苷 y k Substitute the expressions for x and y into the equation of an ellipse. The equation of the ellipse with center at 共h, k兲 is 共x h兲2 共 y k兲2 苷1 a2 b2

6.2 y (h + a, k)

(h − a, k)

379

Ellipses

Standard Forms of the Equation of an Ellipse with Center at (h, k)

Major Axis Parallel to the x-Axis

(h, k)

The standard form of the equation of an ellipse with center at 共h, k兲 and major axis parallel to the x-axis (see Figure 6.26a) is given by

(h + c, k)

(h − c, k)

x

a. Major axis parallel to x-axis

共x h兲2 共 y k兲2 苷 1, a2 b2

ab

The length of the major axis is 2a. The length of the minor axis is 2b. The coordinates of the vertices are 共h a, k兲 and 共h a, k兲, and the coordinates of the foci are 共h c, k兲 and 共h c, k兲, where c2 苷 a2 b2.

y (h, k + a)

Major Axis Parallel to the y-Axis

(h, k + c)

The standard form of the equation of an ellipse with center at 共h, k兲 and major axis parallel to the y-axis (see Figure 6.26b) is given by (h, k)

共x h兲2 共 y k兲2 苷 1, b2 a2

x (h, k − c)

ab

The length of the major axis is 2a. The length of the minor axis is 2b. The coordinates of the vertices are 共h, k a兲 and 共h, k a兲, and the coordinates of the foci are 共h, k c兲 and 共h, k c兲, where c 2 苷 a2 b2.

(h, k − a)

b. Major axis parallel to y-axis

Figure 6.26

EXAMPLE 2

»

Find the Center, Vertices, and Foci of an Ellipse

Find the center, vertices, and foci of the ellipse 4x 2 9y 2 8x 36y 4 苷 0. Sketch the graph.

Solution Write the equation of the ellipse in standard form by completing the square. y

x V2(−2, −2)

V1(4, −2)

C(1, −2)

F2(1 − 5, −2)

F1(1 + 5, −2)

共x 1兲2 共 y 2兲2 苷1 9 4 Figure 6.27

4x 2 9y 2 8x 36y 4 苷 0 4x 2 8x 9y 2 36y 苷 4 4共x 2 2x兲 9共 y 2 4y兲苷 4 4共x 2 2x 1兲 9共 y 2 4y 4兲苷 4 4 36 4共x 1兲2 9共 y 2兲2 苷 36 共x 1兲2 共 y 2兲2 苷1 9 4

• Rearrange terms. • Factor. • Complete the square. • Factor. • Divide each side by 36.

From the equation of the ellipse in standard form, the coordinates of the center of the ellipse are 共1, 2兲. Because the larger denominator is 9, the major axis is parallel to the x-axis and a2 苷 9. Thus a 苷 3. The vertices are 共4, 2兲 and 共2, 2兲. Continued

䉴

380

Chapter 6

Topics in Analytic Geometry To find the coordinates of the foci, we find c. c 2 苷 a 2 b2 苷 9 4 苷 5 c 苷兹5 The foci are 共 1 兹5, 2 兲 and 共 1 兹5, 2 兲. See Figure 6.27.

»

Try Exercise 28, page 385

Integrating Technology A graphing utility can be used to graph an ellipse. For instance, consider the equation 4x 2 9y 2 8x 36y 4 苷 0 from Example 2. Rewrite the equation as 9y 2 36y 共4x 2 8x 4兲苷 0 TO REVIEW

Quadratic Formula See page 5.

In this form, the equation is a quadratic equation in terms of the variable y with A 苷 9, B 苷 36, and C 苷 4x 2 8x 4 Apply the quadratic formula to produce y苷

36 兹1296 36共4x 2 8x 4兲 18

36 兹1296 36共4x 2 8x 4兲 is the part of the 18 ellipse on or above the line y 苷 2 (see Figure 6.28). The graph of Y1 苷

36 兹1296 36共4x 2 8x 4兲 is the part of the 18 ellipse on or below the line y 苷 2, as shown in Figure 6.28. The graph of Y2 苷

3

− 4.7

4.7

Y1

Y2 −6

Figure 6.28

One advantage of this graphing procedure is that it does not require us to write the given equation in standard form. A disadvantage of the graphing procedure is that it does not indicate where the foci of the ellipse are located.

6.2 y

EXAMPLE 3

3

»

Ellipses

381

Find the Equation of an Ellipse

Find the standard form of the equation of the ellipse with center at 共4, 2兲, foci F2共4, 1兲 and F1共4, 5兲, and minor axis of length 10, as shown in Figure 6.29.

F2(4, 1) x

2

Solution C(4, −2)

F1(4, −5) −8

Because the foci are on the major axis, the major axis is parallel to the y-axis. The distance from the center of the ellipse to a focus is c. The distance between the center 共4, 2兲 and the focus 共4, 1兲 is 3. Therefore, c 苷 3. The length of the minor axis is 2b. Thus 2b 苷 10 and b 苷 5. To find a2, use the equation c 2 苷 a2 b2. 9 苷 a2 25 a2 苷 34

Figure 6.29

Thus the equation in standard form is 共x 4兲2 共 y 2兲2 苷1 25 34

»

Try Exercise 44, page 386

Eccentricity of an Ellipse take note Eccentric literally means “out of the center.” Eccentricity is a measure of how much an ellipse is

unlike a set of points the same distance from the center. The higher the eccentricity, the more unlike a circle the ellipse is, and therefore the longer and thinner it

The graph of an ellipse can be very long and thin, or it can be much like a circle. The eccentricity of an ellipse is a measure of its “roundness.” Eccentricity (e) of an Ellipse The eccentricity e of an ellipse is the ratio of c to a, where c is the distance from the center to a focus and a is one-half the length of the major axis. (See Figure 6.30.) That is, c e苷 a

is. A circle is also a conic section. Its standard form is given on e = 0.60

page 25.

e = 0.80 e = 0.92 e = 0.98

Eccentricity 苷 0.87

Figure 6.30

Figure 6.31

Because c a, for an ellipse, 0 e 1. When e ⬇ 0, the graph is almost a circle. When e ⬇ 1, the graph is long and thin. See Figure 6.31.

382

Chapter 6

Topics in Analytic Geometry EXAMPLE 4

»

Find the Eccentricity of an Ellipse

Find the eccentricity of the ellipse given by 8x 2 9y 2 苷 18.

Solution First, write the equation of the ellipse in standard form. Divide each side of the equation by 18. 8x 2 9y 2 苷1 18 18 4x 2 y2 苷1 9 2 x2 y2 苷1 9兾4 2

•

4 1 9 9兾4

The last step is necessary because the standard form of the equation has coefficients of 1 in the numerator. Thus a2 苷

9 4

and

a苷

3 2

c苷

冑

Use the equation c 2 苷 a2 b2 to find c. c2 苷

9 1 2苷 4 4

and

1 1 苷 4 2

Now find the eccentricity. e苷 The eccentricity of the ellipse is

c 1兾2 1 苷苷 a 3兾2 3

1 . 3

»

Try Exercise 50, page 386

Table 6.1 Planet

Eccentricity

Mercury

0.206

Venus

0.007

Earth

0.017

Mars

0.093

Jupiter

0.049

Saturn

0.051

Uranus

0.046

Neptune

0.005

Applications The planets travel around the sun in elliptical orbits. The sun is located at a focus of the orbit. The eccentricities of the orbits for the planets in our solar system are given in Table 6.1.

QUESTION

ANSWER

Which planet has the most nearly circular orbit?

Neptune has the smallest eccentricity, so it is the planet with the most nearly circular orbit.

6.2

Ellipses

383

The terms perihelion and aphelion are used to denote the position of a planet in its orbit around the sun. The perihelion is the point nearest the sun; the aphelion is the point farthest from the sun. See Figure 6.32. The length of the semimajor axis of a planet’s elliptical orbit is called the mean distance of the planet from the sun.

Perihelion

Aphelion Center

Planet

Sun Mean distance

Figure 6.32

EXAMPLE 5

»

Determine an Equation for the Orbit of Earth

Earth has a mean distance of 93 million miles and a perihelion distance of 91.5 million miles. Find an equation for Earth’s orbit.

Solution A mean distance of 93 million miles implies that the length of the semimajor axis of the orbit is a 苷 93 million miles. Earth’s aphelion distance is the length of the major axis less the length of the perihelion distance. Thus Aphelion distance 苷 2共93兲 91.5 苷 94.5 million miles The distance c from the sun to the center of Earth’s orbit is c 苷 aphelion distance 93 苷 94.5 93 苷 1.5 million miles The length b of the semiminor axis of the orbit is b 苷兹a2 c 2 苷兹932 1.52 苷兹8646.75 An equation of Earth’s orbit is x2 y2 苷1 2 93 8646.75

»

Try Exercise 58, page 387

Sound waves, although different from light waves, have a similar reflective property. When sound is reflected from a point P on a surface, the angle of incidence equals the angle of reflection. Applying this principle to a room with an elliptical ceiling results in what are called whispering galleries. These galleries are based on the following theorem.

384

Chapter 6

Topics in Analytic Geometry Reflective Property of an Ellipse

P

β

The lines from the foci to a point on an ellipse make equal angles with the tangent line at that point. See Figure 6.33.

α

F2

F1

The reflective property of an ellipse can be used to show that sound waves, or light waves, that emanate from one focus of an ellipse will be reflected to the other focus. The Statuary Hall in the Capitol Building in Washington, D.C., is a whispering gallery. A person standing at one focus of the elliptical ceiling can whisper and be heard by a person standing at the other focus. John Quincy Adams, while a member of the House of Representatives, was aware of this acoustical phenomenon. He situated his desk at a focus of the elliptical ceiling, which allowed him to eavesdrop on the conversations of his political adversaries, who were located near the other focus.

苷 Figure 6.33

EXAMPLE 6

»

Locate the Foci of a Whispering Gallery

A room 88 feet long is constructed to be a whispering gallery. The room has an elliptical ceiling, as shown in Figure 6.34. If the maximum height of the ceiling is 22 feet, determine where the foci are located.

Solution The length a of the semimajor axis of the elliptical ceiling is 44 feet. The height b of the semiminor axis is 22 feet. Thus y

22 ft –40

c 2 苷 a 2 b2 c 2 苷 442 222 c 苷兹442 222 ⬇ 38.1 feet

Elliptical ceiling of a whispering gallery

10 –20

20 88 ft

Figure 6.34

40 x

The foci are located about 38.1 feet from the center of the elliptical ceiling, along its major axis.

»

Try Exercise 60, page 387

Topics for Discussion 1. In every ellipse, the length of the semimajor axis a is greater than the length of the semiminor axis b and greater than the distance c from a focus to the center of the ellipse. Do you agree? Explain. 2. How many vertices does an ellipse have? 3. Every ellipse has two y-intercepts. Do you agree? Explain. 4. Explain why the eccentricity of every ellipse is a number between 0 and 1.

6.2

Ellipses

385

Exercise Set 6.2 1. Examine the following four equations and the graphs labeled i, ii, iii, and iv. Determine which graph is the graph of each equation. a.

c.

共x 1兲2 共y 2兲2 苷1 9 1

b.

x2 y2 苷1 4 9

i.

d.

−4.7

4.7

iv.

7.2 C(−1, 2)

x2 y2 苷1 49 36

5.

x2 y2 苷1 9 4

6.

x2 y2 苷1 64 25

7.

x2 y2 苷1 9 7

8.

y2 x2 苷1 5 4

9.

4x 2 y2 苷1 9 16

10.

x 2 9y 2 苷1 9 16

−3.1

11.

共x 3兲2 共 y 2兲2 苷1 25 16

12.

共x 3兲2 共 y 1兲2 苷1 9 16

4.6

13.

共x 2兲2 y2 苷1 9 25

14.

共 y 2兲2 x2 苷1 25 81

15.

共x 1兲2 共 y 3兲2 苷1 21 4

16.

共x 5兲2 共 y 3兲2 苷1 9 7

17.

9共x 1兲2 共 y 1兲2 苷1 16 9

18.

共x 6兲2 25y 2 苷1 25 144

C(−1, 2)

−9.4

9.4

−4.7

4.7 −1.6

−5.2

2. Examine the following four equations and the graphs labeled i, ii, iii, and iv. Determine which graph is the graph of each equation. a.

共x 3兲2 共y 2兲2 苷1 1 4

c.

共x 3兲共y 2兲苷1 25 16 2

i.

4.

3.1

−3.1

iii.

x2 y2 苷1 16 25

共x 1兲2 共y 2兲2 苷1 9 16

−4.7

4.7

3. x2 y2 苷1 16 4

ii.

3.1

In Exercises 3 to 34, find the center, vertices, and foci of the ellipse given by each equation. Sketch the graph.

b.

2

2

d.

21. 25x 2 16y 2 苷 400

22. 25x 2 12y 2 苷 300

23. 64x 2 25y 2 苷 400

24. 9x 2 64y 2 苷 144

2

25. 4x 2 y 2 24x 8y 48 苷 0

6.2

26. x 2 9y 2 6x 36y 36 苷 0 27. 5x 2 9y 2 20x 54y 56 苷 0

−9.4

4.7

20. 5x 2 4y 2 苷 20

x y 苷1 4 16

ii.

3.1

−4.7

x2 y2 苷1 9 4

19. 3x 2 4y 2 苷 12

9.4

28. 9x 2 16y 2 36x 16y 104 苷 0 −6.2

−3.1

iii.

iv.

5.1

29. 16x 2 9y 2 64x 80 苷 0 30. 16x 2 9y 2 36y 108 苷 0

8.2 C(3, 2)

31. 25x 2 16y 2 50x 32y 359 苷 0

C(3, 2) 0 −1.1

9.4

−9.4

9.4 −4.2

32. 16x 2 9y 2 64x 54y 1 苷 0

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33. 8x 2 25y 2 48x 50y 47 苷 0

50. Foci at 共0, 3兲 and 共0, 3兲, eccentricity

34. 4x 2 9y 2 24x 18y 44 苷 0

1 4

51. Eccentricity

2 , foci at 共1, 3兲 and 共3, 3兲 5

In Exercises 35 to 46, find the equation in standard form of each ellipse, given the information provided.

52. Eccentricity

1 , foci at 共2, 4兲 and 共2, 2兲 4

35. Center 共0, 0兲, major axis of length 10, foci at 共4, 0兲 and 共4, 0兲

53. Eccentricity

2 , major axis of length 24 on the y-axis, center 3

36. Center 共0, 0兲, minor axis of length 6, foci at 共0, 4兲 and 共0, 4兲

at 共0, 0兲

37. Vertices 共6, 0兲, 共6, 0兲; ellipse passes through 共0, 4兲 and 共0, 4兲

54. Eccentricity

38. Vertices 共7, 0兲, 共7, 0兲; ellipse passes through 共0, 5兲 and 共0, 5兲

55.

39. Major axis of length 12 on the x-axis, center at 共0, 0兲; ellipse passes through 共2, 3兲 40. Major axis of length 8, center at 共0, 0兲; ellipse passes through 共2, 2兲 41. Center 共2, 4兲, vertices 共6, 4兲 and 共2, 4兲, foci at 共5, 4兲 and 共1, 4兲

at 共0, 0兲

3 , major axis of length 15 on the x-axis, center 5

MEDICINE A lithotripter is an instrument used to remove a kidney stone in a patient without having to do surgery. A high-frequency sound wave is emitted from a source that is located at the focus of an ellipse. The patient is placed so that the kidney stone is located at the other focus of the ellipse. If the equation of the ellipse is 共x 11兲2 y2 苷 1 (x and y are measured in centimeters), 484 64 where, to the nearest centimeter, should the patient’s kidney stone be placed so that the reflected sound hits the kidney stone?

42. Center 共0, 3兲, minor axis of length 4, foci at 共0, 0兲 and 共0, 6兲 43. Center 共2, 4兲, major axis parallel to the y-axis and of length 10; ellipse passes through the point 共3, 3兲

−

+

Ultrasound emitter

Kidney

44. Center 共4, 1兲, minor axis parallel to the y-axis and of length 8; ellipse passes through the point 共0, 4兲 45. Vertices 共5, 6兲 and 共5, 4兲, foci at 共5, 4兲 and 共5, 2兲

Elliptic reflector

Kidney stone

46. Vertices 共7, 1兲 and 共5, 1兲, foci at 共5, 1兲 and 共3, 1兲

In Exercises 47 to 54, use the eccentricity of each ellipse to find its equation in standard form.

2 , major axis on the x-axis and of length 10, 5 center at 共0, 0兲

47. Eccentricity

48. Eccentricity

56. CONSTRUCTION A circular vent pipe is placed on a roof that has a slope of 4 , as shown in the figure 5 at the right.

3 , foci at 共9, 0兲 and 共9, 0兲 4

a. Use the slope to find the value of h.

2 3

b. The intersection of the vent pipe and the roof

49. Foci at 共0, 4兲 and 共0, 4兲, eccentricity

h

4.5 in.

6.2 is an ellipse. To the nearest hundredth of an inch, what are the lengths of the major and minor axes?

Elliptical ceiling of a whispering gallery

387

Ellipses

y

c. Find an equation of the ellipse that should be cut from the roof so that the pipe will fit. h

57.

THE ORBIT OF SATURN The distance from Saturn to the sun at Saturn’s aphelion is 934.34 million miles, and the distance from Saturn to the sun at its perihelion is 835.14 million miles. Find an equation for the orbit of Saturn.

Foci

–50

32 ft

50

x

100 ft

If the foci are to be located 32 feet to the right and the left of center, find the height h of the elliptical ceiling (to the nearest tenth of a foot). Perihelion 835.14 million miles

Aphelion 934.34 million miles

61.

Sun

58.

y

THE ORBIT OF VENUS Venus has a mean distance from the sun of 67.08 million miles, and the distance from Venus to the sun at its aphelion is 67.58 million miles. Find an equation for the orbit of Venus.

9 AU x

59. WHISPERING GALLERY An architect wishes to design a large room that will be a whispering gallery. The ceiling of the room has a cross section that is an ellipse, as shown in the following figure. y

20 30 ft

– 50

36 AU

62.

Elliptical ceiling of a whispering gallery

50

x

100 ft

How far to the right and left of center are the foci located? 60. WHISPERING GALLERY An architect wishes to design a large room 100 feet long that will be a whispering gallery. The ceiling of the room has a cross section that is an ellipse, as shown in the following figure.

HALLEY’S COMET Find the equation of the path of Halley’s comet in astronomical units by letting one focus (the sun) be at the origin and letting the other focus be on the positive x-axis. The length of the major axis of the orbit of Halley’s comet is approximately 36 astronomical units (36 AU), and the length of the minor axis is 9 AU 共1 AU 苷 92,960,000 miles兲.

ELLIPTICAL POOL TABLE A pool table in the shape of an ellipse has only one pocket, which is located at a focus of the ellipse. A cue ball is placed at the other focus of the ellipse. Striking the cue ball firmly in any direction causes it to go into the pocket (assuming no side or back spin is introduced to the motion of the cue ball). Explain why this happens.

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63. BRIDGE CLEARANCE During the 1960s, the London Bridge was dismantled and shipped to Lake Havasu City, Arizona, where it was reconstructed. Use the following diagram of an arch of the bridge to determine the vertical distance h from the water to the arch 55 feet to the right of point O, which is the center of the semielliptical arch. The distance from O to the water is 1 foot. Round to the nearest foot.

65. CONSTRUCTION A carpenter needs to cut a semielliptical form from a 3-foot by 8-foot sheet of plywood, as shown in the following diagram. a. Where should the carpenter place the push pins? b. How long is the string that connects the push pins?

3 ft O 4 ft

34 ft water level

1 ft

O

55 ft

8 ft

h

66. 150 ft

64. ELLIPTICAL GEARS The figure below shows two elliptical gears. Search the Internet for an animation that demonstrates the motion of an elliptical gear that is driven by another elliptical gear rotating at a constant speed. Explain what happens to the gear on the left as the gear on the right rotates once at a constant angular speed.

ORBIT OF MARS Mars travels around the sun in an elliptical orbit with the sun located at a focus of the orbit. The orbit has a major axis of 3.04 AU and a minor axis of 2.99 AU. (1 AU is 1 astronomical unit, or approximately 92,960,000 miles, the average distance of Earth from the sun.) Estimate, to the nearest million miles, the perimeter of the orbit of Mars. [Note: The approximate perimeter of an ellipse with semimajor axis a and semiminor axis b is P 苷 p 兹2共a2 b2 兲.]

In Exercises 67 to 69, use the quadratic formula to solve for y in terms of x. Then use a graphing utility to graph each equation.

67. 16x 2 9y 2 36y 108 苷 0 68. 8x 2 25y 2 48x 50y 47 苷 0 Source: http://www.stirlingsouth.com/richard/ develop/engines_under_development.htm

69. 4x 2 9y 2 24x 18y 44 苷 0

Connecting Concepts 70. Explain why the graph of 4x 2 9y 16x 2 苷 0 is or is not an ellipse. Sketch the graph of this equation.

72. Find the equation of the ellipse with foci at 共0, 4兲 and 9 共0, 4兲 that passes through the point ,4 . 5

In Exercises 71 to 74, find the equation in standard form of each ellipse by using the definition of an ellipse.

73. Find the equation of the ellipse with foci at 共1, 2兲 and 共3, 2兲 that passes through the point 共3, 5兲.

71. Find the equation of the ellipse with foci at 共3, 0兲 and 9 . 共3, 0兲 that passes through the point 3, 2

冉冊

冉冊

74. Find the equation of the ellipse with foci at 共1, 1兲 and 3 共1, 7兲 that passes through the point ,1 . 4

冉冊

6.2 In Exercises 75 and 76, find the latus rectum of the given ellipse. A line segment with endpoints on the ellipse that is perpendicular to the major axis and passes through a focus is a latus rectum of the ellipse.

389

Ellipses

75. Find the length of a latus rectum of the ellipse given by 共x 1兲2 共 y 1兲2 苷1 9 16 76. Find the length of a latus rectum of the ellipse given by 9x 2 16y 2 36x 96y 36 苷 0

Latus rectum F1

F2

77. Show that for any ellipse, the length of a latus rectum 2b2 is . a 78. Use the definition of an ellipse to find the equation of an ellipse with center at 共0, 0兲 and foci at 共0, c兲 and 共0, c兲.

Projects 1.

KEPLER’S LAWS The German astronomer Johannes Kepler (1571 – 1630) derived three laws that describe how the planets orbit the sun. Write an essay that includes biographical information about Kepler and a statement of Kepler’s Laws. In addition, use Kepler’s Laws to answer the following questions. a. Where is a planet located in its orbit around the sun when it achieves its greatest velocity? b. What is the period of Mars if it has a mean distance from the sun of 1.52 astronomical units? (Hint: Use Earth as a reference with a period of 1 year and a mean distance from the sun of 1 astronomical unit.)

2.

3.

NEPTUNE The position of the planet Neptune was discovered by using celestial mechanics and mathematics. Write an essay that tells how, when, and by whom Neptune was discovered. GRAPH THE COLOSSEUM Some of the Colosseum scenes in the movie Gladiator (Universal Studios, 2000) were computer-generated.

a. You can create a simple but accurate scale image of the exterior of the Colosseum by using a computer and the mathematics software program Maple. Open a new Maple worksheet and enter the following two commands. with(plots); plots[implicitplot3d]((x^2)/(307.5^2)+(y^2)/(255^2)= 1, x= –310..310, y= –260..260, z=0..157, scaling=CONSTRAINED, style=PATCHNOGRID, axes=FRAMED); Execute each of the commands by placing the cursor in a command and pressing the ENTER key. After execution of the second command, a three-dimensional graph will appear. Click and drag on the graph to rotate the image. b. A graphing calculator can also be used to generate a simple “graph” of the exterior of the Colosseum. Here is a procedure for the TI-83/TI-83 Plus/TI-84 Plus graphing calculator. Enter the following in the WINDOW menu. Xmin=–4.7 Xmax=4.7 Ymax=9 Yscl=1

Xscl=1

Ymin=–4

Enter the following equations in the Y苷 menu. Y1=兹6(9–X2) Press:

Y2=Y1+4 Y3=–Y1 Y4=Y3+4

QUIT (2nd MODE)

Enter: Shade(Y3,Y4) and press ENTER. Note: “Shade(” is in the DRAW menu. Explain why this “Colosseum graph” appears to be constructed with ellipses even though the functions entered in the Y苷 menu are the equations of semicircles.

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Section 6.3 ■

■

■

■

Hyperbolas with Center at (0, 0) Hyperbolas with Center at (h, k) Eccentricity of a Hyperbola Applications

Hyperbolas P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A27.

PS1. Find the midpoint and the length of the line segment between P1共4, 3兲 and P2共2, 1兲. [1.2] PS2. Solve: 共x 1兲共x 3兲苷 5 [1.1] PS3. Simplify:

4 兹8

PS4. Complete the square of 4x 2 24x and write the result as the square of a binomial. [1.2] PS5. Solve

x2 y2 苷 1 for y. [1.2] 4 9

PS6. Graph:

take note If a plane intersects a cone along the axis of the cone, the resulting curve is two intersecting straight lines. This is the degenerate form of a hyperbola. See the accompanying figure.

共x 2兲2 共 y 3兲2 苷 1 [6.2] 16 9

A hyperbola is a conic section formed when a plane intersects a right circular cone at a certain angle. If b is the angle at which the plane intersects the axis of the cone and a is the angle shown in Figure 6.35, a hyperbola is formed when 0 b a or when the plane is parallel to the axis of the cone. As with the other conic sections, there is a definition of a hyperbola in terms of a certain set of points in the plane.

Axis

Axis

β

β

α

α

l

l

Figure 6.35

Definition of a Hyperbola A hyperbola is the set of all points in the plane the difference between whose distances from two fixed points (foci) is a positive constant.

Degenerate hyperbola

This definition differs from that of an ellipse in that the ellipse was defined in terms of the sum of two distances, whereas the hyperbola is defined in terms of the difference of two distances.

6.3 Conjugate axis

(0, b) V1(a, 0)

F2(−c, 0)

x

F1(c, 0)

V2(−a, 0)

(0, −b)

Figure 6.36 y

V1(a, 0)

F2(− c, 0)

The transverse axis of a hyperbola shown in Figure 6.36 is the line segment joining the intercepts. The midpoint of the transverse axis is called the center of the hyperbola. The conjugate axis is a line segment that passes through the center of the hyperbola and is perpendicular to the transverse axis. The length of the transverse axis is customarily represented as 2a, and the distance between the two foci is represented as 2c. The length of the conjugate axis is represented as 2b. The vertices of a hyperbola are the points where the hyperbola intersects the transverse axis. To determine the positive constant stated in the definition of a hyperbola, consider the point V1共a, 0兲, which is one vertex of a hyperbola, and the points F1共c, 0兲 and F2共c, 0兲, which are the foci of the hyperbola (see Figure 6.37). The difference between the distance from V1共a, 0兲 to F1共c, 0兲, c a, and the distance from V1共a, 0兲 to F2共c, 0兲, c a, must be a constant. By subtracting these distances, we find 兩共c a兲共c a兲兩苷兩2a兩苷 2a

c−a

x

F1(c, 0)

c+a

391

Hyperbolas with Center at (0, 0)

y

Transverse axis

Hyperbolas

Thus the constant is 2a, and it is the length of the transverse axis. The absolute value is used to ensure that the distance is a positive number.

Standard Forms of the Equation of a Hyperbola with Center at the Origin Figure 6.37

Transverse Axis on the x-Axis The standard form of the equation of a hyperbola with center at the origin and transverse axis on the x-axis (see Figure 6.38a) is given by

y

(− a, 0)

x2 y2 苷1 a2 b2

(a, 0) (c, 0)

(−c, 0)

x

The coordinates of the vertices are 共a, 0兲 and 共a, 0兲, and the coordinates of the foci are 共c, 0兲 and 共c, 0兲, where c2 苷 a2 b2.

Transverse Axis on the y-Axis

a. Transverse axis on the x-axis

The standard form of the equation of a hyperbola with center at the origin and transverse axis on the y-axis (see Figure 6.38b) is given by

y (0, c)

y2 x2 苷1 2 a b2

(0, a) x (0, −a)

The coordinates of the vertices are 共0, a兲 and 共0, a兲, and the coordinates of the foci are 共0, c兲 and 共0, c兲, where c 2 苷 a2 b2.

(0, −c)

b. Transverse axis on the y-axis

Figure 6.38

By looking at the equations, it is possible to determine the location of the transverse axis by finding which term in the equation is positive. When the x 2 term is positive, the transverse axis is on the x-axis. When the y 2 term is positive, the transverse axis is on the y-axis.

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QUESTION y 4

(5, 0) (−4, 0)

y2 x2 苷 1. Because the x 2 16 9 term is positive, the transverse axis is on the x-axis, a2 苷 16, and thus a 苷 4. The vertices are 共4, 0兲 and 共4, 0兲. To find the foci, we determine c. Consider the hyperbola given by the equation

(− 5, 0) –8

y2 x2 苷 1, is the transverse axis on the x-axis For the graph of 9 4 or the y-axis?

(4, 0)

8

x

c 2 苷 a2 b2 苷 16 9 苷 25 c 苷兹25 苷 5

–4

x2 y2 苷1 16 9 Figure 6.39

The foci are 共5, 0兲 and 共5, 0兲. The graph is shown in Figure 6.39. Each hyperbola has two asymptotes that pass through the center of the hyperbola. The asymptotes of the hyperbola are a useful guide to sketching the graph of the hyperbola. Asymptotes of a Hyperbola with Center at the Origin The asymptotes of the hyperbola defined by equations y 苷

b b x and y 苷 x (see Figure 6.40a). a a

The asymptotes of the hyperbola defined by equations y 苷

x2 y2 苷 1 are given by the 2 a b2

y2 x2 苷 1 are given by the a2 b2

a a x and y 苷 x (see Figure 6.40b). b b

y=

y

b x a y=

a x b

y=−

a x b

a

b

b

a x

y=−

b x a

a. Asymptotes of

x2 y2 − 2 =1 2 a b

b. Asymptotes of

y2 x2 − 2 =1 2 a b

Figure 6.40

One method for remembering the equations of the asymptotes is to write the equation of a hyperbola in standard form, then replace 1 by 0 and solve for y. x2 y2 b2 2 b 2 苷 0 so y 苷 x 2 2 2 x , or y 苷

a b a a x2 a2 2 a y2 2 苷 0 so y 苷 x 2 2 2 x , or y 苷

a b b b ANSWER

Because the y 2-term is positive, the transverse axis is on the y-axis.

6.3 EXAMPLE 1

»

393

Hyperbolas

Find the Vertices, Foci, and Asymptotes of a Hyperbola

Find the vertices, foci, and asymptotes of the hyperbola given by the y2 x2 equation 苷 1. Sketch the graph. 9 4 y

F1(0,

Solution

13)

4 y=

3 x 2

Because the y 2 term is positive, the transverse axis is on the y-axis. We know that a2 苷 9; thus a 苷 3. The vertices are V1共0, 3兲 and V2共0, 3兲. c 2 苷 a 2 b2 苷 9 4 c 苷兹13

V1(0, 3)

−3

3

x

y=−

3 x 2

V2(0, − 3)

−4 F2(0, − 13)

y2 x2 苷1 9 4 Figure 6.41

The foci are F1共 0, 兹13 兲 and F2共 0, 兹13 兲. Because a 苷 3 and b 苷 2 (b2 苷 4), the equations of the asymptotes are 3 3 y 苷 x and y 苷 x. 2 2 To sketch the graph, we draw a rectangle that has its center at the origin and has dimensions equal to the lengths of the transverse and conjugate axes. The asymptotes are extensions of the diagonals of the rectangle. See Figure 6.41.

»

Try Exercise 6, page 399

Hyperbolas with Center at (h, k) Using a translation of coordinates similar to that used for ellipses, we can write the equation of a hyperbola with center at the point 共h, k兲. Given coordinate axes labeled x and y , an equation of a hyperbola with center at the origin is 共x 兲2 共 y 兲2 2 苷1 a2 b

(1)

Now place the origin of this coordinate system at the point 共h, k兲 of the xy-coordinate system, as shown in Figure 6.42. The relationship between an ordered pair in the x y -coordinate system and one in the xy-coordinate system is given by the transformation equations x 苷 x h y 苷 y k Substitute the expressions for x and y into Equation (1). The equation of a hyperbola with center at 共h, k兲 is 共x h兲2 共 y k兲2 苷1 a2 b2

y

y'

(x, y)

(x', y')

(h, k) x' x

Figure 6.42

394

Chapter 6

y

y−k=

Topics in Analytic Geometry

b (x − h) a

Standard Forms of the Equation of a Hyperbola with Center at (h, k)

Transverse Axis Parallel to the x-Axis V2(h − a, k)

V1(h + a, k)

(h, k)

F2(h − c, k)

F1(h + c, k)

x y−k=−

b (x − h) a

a. Transverse axis parallel to the x-axis y

共x h兲2 共 y k兲2 苷1 a2 b2 The coordinates of the vertices are V1共h a, k兲 and V2共h a, k兲. The coordinates of the foci are F1共h c, k兲 and F2共h c, k兲, where c2 苷 a2 b2. b The equations of the asymptotes are y k 苷共x h兲. a

Transverse Axis Parallel to the y-Axis F1(h, k + c)

V1(h, k + a) V2(h, k − a)

The standard form of the equation of a hyperbola with center at 共h, k兲 and transverse axis parallel to the x-axis (see Figure 6.43a) is given by

The standard form of the equation of a hyperbola with center at 共h, k兲 and transverse axis parallel to the y-axis (see Figure 6.43b) is given by y−k=

(h, k)

x F2(h, k − c) y−k=−

共 y k兲2 共x h兲2 苷1 a2 b2

a (x − h) b

a (x − h) b

The coordinates of the vertices are V1共h, k a兲 and V2共h, k a兲. The coordinates of the foci are F1共h, k c兲 and F2共h, k c兲, where c2 苷 a2 b2. The equations of the asymptotes are y k 苷

a 共x h兲. b

b. Transverse axis parallel to the y-axis

Figure 6.43

EXAMPLE 2

»

Find the Center, Vertices, Foci, and Asymptotes of a Hyperbola

Find the center, vertices, foci, and asymptotes of the hyperbola given by the equation 4x 2 9y 2 16x 54y 29 苷 0. Sketch the graph.

Solution Write the equation of the hyperbola in standard form by completing the square. 4x 2 9y 2 16x 54y 29 苷 0 4x 2 16x 9y 2 54y 苷 29 4共x 2 4x兲 9共 y 2 6y兲苷 29 4共x 2 4x 4兲 9共 y 2 6y 9兲苷 29 16 81 4共x 2兲2 9共 y 3兲2 苷 36 共 y 3兲2 共x 2兲2 苷1 4 9

• Rearrange terms. • Factor. • Complete the square. • Factor. • Divide each side by 36.

6.3 y 8

F1(2, 3 +

13)

c 2 苷 a 2 b2 苷 4 9 c 苷兹13

V2(2, 1)

F2(2, 3 –

395

The coordinates of the center are 共2, 3兲. Because the term containing 共 y 3兲2 is positive, the transverse axis is parallel to the y-axis. We know that a2 苷 4; thus a 苷 2. The vertices are 共2, 5兲 and 共2, 1兲. See Figure 6.44. To find the coordinates of the foci, we find c.

V1(2, 5)

−2

Hyperbolas

13)

共 y 3兲2 共x 2兲2 苷1 4 9

8

x

The foci are 共 2, 3 兹13 兲 and 共 2, 3 兹13 兲. We know that b2 苷 9; thus 2 b 苷 3. The equations of the asymptotes are y 3 苷

共x 2兲, which 3 simplifies to

冉冊

y苷

Figure 6.44

2 5 x 3 3

and

y苷

13 2 x 3 3

»

Try Exercise 28, page 399

Integrating Technology A graphing utility can be used to graph a hyperbola. For instance, consider the equation 4x 2 9y 2 16x 54y 29 苷 0 from Example 2. Rewrite the equation as 9y 2 54y 共4x 2 16x 29兲苷 0 In this form, the equation is a quadratic equation in terms of the variable y with A 苷 9, B 苷 54, and C 苷 4x 2 16x 29 Apply the quadratic formula to produce y苷

54 兹2916 36共4x 2 16x 29兲 18

54 兹2916 36共4x 2 16x 29兲 is the upper 18 branch of the hyperbola (see Figure 6.45). The graph of Y1 苷

54 兹2916 36共4x 2 16x 29兲 is the lower 18 branch of the hyperbola, as shown in Figure 6.45. One advantage of this graphing 10 procedure is that it does not require us to write the given equation in standard form. A disadvantage of the graphing procedure is that it does not indicate where the foci of the hyperbola are located.

The graph of Y2 苷

−4

−2

Figure 6.45

9

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Chapter 6

Topics in Analytic Geometry

Eccentricity of a Hyperbola

e = 2.23

e = 1.41

e = 1.1

e = 1.03

The graph of a hyperbola can be very wide or very narrow. The eccentricity of a hyperbola is a measure of its “wideness.” Definition of the Eccentricity (e) of a Hyperbola The eccentricity e of a hyperbola is the ratio of c to a, where c is the distance from the center to a focus and a is half the length of the transverse axis. e苷

c a

For a hyperbola, c a and therefore e 1. As the eccentricity of the hyperbola increases, the graph becomes wider and wider, as shown in Figure 6.46.

Figure 6.46

EXAMPLE 3

»

Find the Equation of a Hyperbola Given Its Eccentricity

Find the standard form of the equation of the hyperbola that has eccentricity 3 , center at the origin, and a focus at 共6, 0兲. 2

Solution Because the focus is located at 共6, 0兲 and the center is at the origin, c 苷 6. An extension of the transverse axis contains the foci, so the transverse axis is on the x-axis. y

e苷

8

8 x

−8

3 c 苷 2 a 3 6 苷 2 a a苷4

• Substitute 6 for c. • Solve for a.

To find b , use the equation c 苷 a b2 and the values for c and a. 2

2

c 2 苷 a 2 b2 36 苷 16 b2 b2 苷 20

−8

x2 y2 苷1 16 20 Figure 6.47

2

The equation of the hyperbola is

x2 y2 苷 1. See Figure 6.47. 16 20

»

Try Exercise 50, page 400

Applications Orbits of Comets In Section 6.2 we noted that the orbits of the planets are elliptical. Some comets have elliptical orbits also, the most notable being Halley’s comet, whose eccentricity is 0.97. See Figure 6.48.

6.3

Math Matters

397

Hyperbolas

Earth Sun

Orbit of Halley's comet

Venus

Mars Neptune Jupiter Uranus Saturn Not drawn to scale.

Figure 6.48

Other comets have hyperbolic orbits with the sun at a focus. These comets pass by the sun only once. The velocity of a comet determines whether its orbit is elliptical or hyperbolic. Caroline Herschel (1750–1848) became interested in mathematics and astronomy after her brother William discovered the planet Uranus. She was the first woman to receive credit for the discovery of a comet. In fact, between 1786 and 1797, she discovered eight comets. In 1828 she completed a catalog of over 2000 nebulae, a feat for which the Royal Astronomical Society of England presented her with its prestigious gold medal.

Hyperbolas as an Aid to Navigation Consider two radio transmitters, T1 and T2 , placed some distance apart. A ship with electronic equipment measures the difference between the times it takes signals from the transmitters to reach the ship. Because the difference between the times is proportional to the difference between the distances of the ship from the transmitters, the ship must be located on the hyperbola with foci at the two transmitters. Using a third transmitter, T3, we can find a second hyperbola with foci T2 and T3. The ship lies on the intersection of the two hyperbolas, as shown in Figure 6.49.

T2

T1

T3

Figure 6.49 y 400 N

EXAMPLE 4

Ship 300

200 Ocean 100 T1 –200

–100

100

200

a.

Find an equation of a hyperbola (with foci located at T1 and T2 ) on which the ship lies. See Figure 6.50. (Assume the radio signals travel at 0.186 mile per microsecond.)

b.

If the ship is directly north of transmitter T1, determine how far (to the nearest mile) the ship is from the transmitter.

x

Land

Figure 6.50

Determine the Position of a Ship

Two radio transmitters are positioned along a coastline, 500 miles apart. See Figure 6.50. Using a LORAN (LOng RAnge Navigation) system, a ship determines that a radio signal from transmitter T1 reaches the ship 1600 microseconds before it receives a simultaneous signal from transmitter T2.

B

T2

»

Continued

䉴

398

Chapter 6

Topics in Analytic Geometry Solution The ship lies on a hyperbola at point B, with foci at T1 and T2. The difference of the distances d共T2 , B兲 and d共T1, B兲 is given by

a.

Distance 苷 rate time 苷 0.186 mile兾microsecond 1600 microseconds 苷 297.6 mile This indicates that the ship is located on a hyperbola with transverse axis of length 297.6 miles. Thus 2a 苷 297.6 miles and a 苷 148.8 miles. Figure 6.50 shows that the foci are located at 共250, 0兲 and 共250, 0兲. Thus c 苷 250 miles, and b 苷兹c 2 a2 苷兹250 2 148.82 ⬇ 200.9 miles The ship is located on the hyperbola given by x2 y2 苷1 2 148.8 200.9 2 If the ship is directly north of T1, then x 苷 250, and the distance from the ship to the transmitter T1 is y, where

b.

y α

The ship is about 271 miles north of transmitter T1.

»

Try Exercise 56, page 400

P(x, y) β

F1(−c, 0)

F2(c, 0)

y2 250 2 苷 1 200.9 2 148.82 200.9 y苷兹250 2 148.82 ⬇ 271 miles 148.8

x

Hyperbolas also have a reflective property that makes them useful in many applications.

Reflective Property of a Hyperbola a苷b

Figure 6.51

The lines from the foci to a point on a hyperbola make equal angles with the tangent line at that point. See Figure 6.51.

Topics for Discussion 1. In every hyperbola, the distance c from a focus to the center of the hyperbola is greater than half the length of the transverse axis. Do you agree? Explain. 2. How many vertices does a hyperbola have? 3. Explain why the eccentricity of every hyperbola is a number greater than 1. 4. Is the conjugate axis of a hyperbola perpendicular to the transverse axis of the hyperbola?

6.3

Hyperbolas

399

Exercise Set 6.3 1. Examine the following four equations and the graphs labeled i, ii, iii, and iv. Determine which graph is the graph of each equation.

In Exercises 3 to 28, find the center, vertices, foci, and asymptotes for the hyperbola given by each equation. Graph each equation.

3.

x2 y2 苷1 16 25

4.

y2 x2 苷1 16 9

5.

y2 x2 苷1 4 25

6.

y2 x2 苷1 25 36

7.

x2 y2 苷1 7 9

8.

y2 x2 苷1 5 4

9.

4x 2 y 2 苷1 9 16

10.

x 2 9y 2 苷1 9 16

11.

共x 3兲2 共 y 4兲2 苷1 16 9

12.

y2 共x 3兲2 苷1 25 4

13.

共 y 2兲2 共x 1兲2 苷1 4 16

14.

共 y 2兲2 共x 1兲2 苷1 36 49

15.

共x 2兲2 y 2 苷1 9 25

16.

x 2 共 y 2兲2 苷1 25 81

2. Examine the following four equations and the graphs labeled i, ii, iii, and iv. Determine which graph is the graph of each equation.

17.

9共x 1兲2 共 y 1兲2 苷1 16 9 共x 6兲2 25y 2 苷1 25 144

共x 1兲2 共y 2兲2 a. 苷1 9 16 c.

x2 y2 b. 苷1 9 4

y2 x2 苷1 4 9

i.

共y 2兲2 共x 1兲2 苷1 4 9

d. ii.

6.2

−9.4

6.2

−9.4

9.4

9.4

−6.2

−6.2

iii.

iv.

8.2

8.2

C(1, 2 −9.4

−9.4

9.4

9.4 −4.2

−4.2

a.

x2 y2 苷1 25 4

b.

共y 1兲2 共x 4兲2 苷1 9 36

18.

c.

共x 4兲2 共y 1兲2 苷1 36 9

d.

y2 x2 苷 1 4

19. x 2 y 2 苷 9 20. 4x 2 y 2 苷 16

i.

ii.

5.7

6.2

21. 16y 2 9x 2 苷 144

−6.4

C(4, −1)

12.4

−9.4

9.4

23. 9y 2 36x 2 苷 4

−6.2

−6.7

22. 9y 2 25x 2 苷 225

24. 16x 2 25y 2 苷 9 iii.

iv.

6.2

−9.4

9.4

−6.2

5.7

−5.4

25. x 2 y 2 6x 8y 3 苷 0 13.4

−6.7

26. 4x 2 25y 2 16x 50y 109 苷 0 27. 9x 2 4y 2 36x 8y 68 苷 0 28. 16x 2 9y 2 32x 54y 79 苷 0

400

Chapter 6

Topics in Analytic Geometry

In Exercises 29 to 34, use the quadratic formula to solve for y in terms of x. Then use a graphing utility to graph each equation.

29. 4x 2 y 2 32x 6y 39 苷 0

In Exercises 49 to 54, use the eccentricity to find the equation in standard form of each hyperbola.

30. x 2 16y 2 8x 64y 16 苷 0

49. Vertices 共1, 6兲 and 共1, 8兲, eccentricity 2

31. 9x 2 16y 2 36x 64y 116 苷 0

50. Vertices (2, 3) and 共2, 3兲, eccentricity

32. 2x 2 9y 2 12x 18y 18 苷 0 33. 4x 2 9y 2 8x 18y 6 苷 0

52. Eccentricity

In Exercises 35 to 48, find the equation in standard form of the hyperbola that satisfies the stated conditions.

35. Vertices 共3, 0兲 and 共3, 0兲, foci 共4, 0兲 and 共4, 0兲 36. Vertices 共0, 2兲 and 共0, 2兲, foci 共0, 3兲 and 共0, 3兲 37. Foci 共0, 5兲 and 共0, 5兲, asymptotes y 苷 2x and y 苷 2x 38. Foci 共4, 0兲 and 共4, 0兲, asymptotes y 苷 x and y 苷 x 39. Vertices 共0, 3兲 and 共0, 3兲, passing through 共2, 4兲 40. Vertices 共5, 0兲 and 共5, 0兲, passing through 共1, 3兲

共0, 4兲 42. Asymptotes y 苷共6, 0兲

5 2

51. Eccentricity 2, foci 共4, 0兲 and 共4, 0兲

34. 2x 2 9y 2 8x 36y 46 苷 0

41. Asymptotes y 苷

48. Passing through 共6, 1兲, slope of an asymptote 2, center 共3, 3兲, transverse axis parallel to the x-axis

1 1 x and y 苷 x, vertices 共0, 4兲 and 2 2

4 , foci 共0, 6兲 and 共0, 6兲 3

4 53. Center 共4, 1兲, conjugate axis of length 4, eccentricity 3 (Hint: There are two answers.) 54. Center 共3, 3兲, conjugate axis of length 6, eccentricity 2 (Hint: There are two answers.) 55. LORAN Two radio transmitters are positioned along the coast, 250 miles apart. A signal is sent simultaneously from each transmitter. The signal from transmitter T2 is received by a ship’s LORAN 500 microseconds after the ship receives the signal from T1 . The radio signals travel at 0.186 mile per microsecond. a. Find an equation of a hyperbola, with foci at T1 and T2 , on which the ship is located. b. If the ship is 100 miles east of the y-axis, determine its distance from the coastline (to the nearest mile). y 400

2 2 x and y 苷 x, vertices 共6, 0兲 and 3 3

N 300 Ship

43. Vertices 共6, 3兲 and 共2, 3兲, foci 共7, 3兲 and 共1, 3兲

200

44. Vertices 共1, 5兲 and 共1, 1兲, foci 共1, 7兲 and 共1, 3兲 45. Foci 共1, 2兲 and 共7, 2兲, slope of an asymptote

5 4

46. Foci 共3, 6兲 and 共3, 2兲, slope of an asymptote 1 47. Passing through 共9, 4兲, slope of an asymptote 共7, 2兲, transverse axis parallel to the y-axis

B

Ocean

1 , center 2

100 T2

T1 –100

–50

50

100

x

250 mi Land

56. LORAN Two radio transmitters are positioned along the coast, 300 miles apart. A signal is sent simultaneously from each transmitter. The signal from transmitter T1 is

6.3 received by a ship’s LORAN 800 microseconds after the ship receives the signal from T2 . The radio signals travel at 0.186 mile per microsecond.

401

Hyperbolas

where x and y are measured in feet. The horizontal cross sections of the tower are circles.

a. Find an equation of a hyperbola, with foci at T1 and T2 , on which the ship is located.

y

b. If the ship continues to travel so that the difference of 800 microseconds is maintained, determine the point at which the ship will reach the coastline.

380 ft

x

y 200 N

a. Find the radius of the top and the base of the tower. Round to the nearest foot.

150

Ship

b. What is the smallest radius of a horizontal cross section?

100 Ocean 50

T2

–90

59. HYPERBOLIC GEAR The following diagram shows a cylindrical worm gear driving a hyperbolic gear. T1 x

90 300 mi Land

57.

hyperbolic gear

SONIC BOOMS When a plane exceeds the speed of sound, a sonic boom is produced by the wake of the sound waves. For a plane flying at 10,000 feet, the circular wave front can be given by worm gear

y 2 苷 x 2 共z 10,000兲2 where z is the height of the wave front above Earth. See the diagram below. Note that the xy-plane is Earth’s surface, which is approximately flat over small distances.

Source: http://www.zakgear.com/Wormoid.html

A center vertical cross section of the hyperbolic gear is shown below. The dimensions given are in inches. y

Plane

y = 0.6

F

G

z

x

y Earth

V2 (−2, 0)

V1(2, 0)

x

y = −0.6

Find and name the equation formed when the wave front hits Earth.

The vertical cross section of a hyperbolic gear

58. COOLING TOWER A vertical cross section of a cooling tower is a portion of a hyperbola, as shown in the following diagram. The standard form of the equation of the hyperbolic cross section is

The eccentricity of the hyperbolic cross section is

共y 220兲2 x2 苷 1, 0 y 380 2 80 180 2

兹17 . 4

a. What is the equation in standard form of the hyperbolic cross section? b. Find the length of diameter FG. Round to the nearest hundredth of an inch.

402

Chapter 6

Topics in Analytic Geometry

60. WATER WAVES If two pebbles are dropped into a pond at different places F1共2, 0兲 and F2共2, 0兲, circular waves are propagated with F1 as the center of one set of circular waves (in green) and F2 as the center of the other set of circular waves (in red).

b. What is the equation of the curve in a.? In Exercises 61 to 68, identify the graph of each equation as a parabola, an ellipse, or a hyperbola. Graph each equation.

61. 4x 2 9y 2 16x 36y 16 苷 0

y

62. 2x 2 3y 8x 2 苷 0 P

63. 5x 4y 2 24y 11 苷 0 64. 9x 2 25y 2 18x 50y 苷 0

x

65. x 2 2y 8x 苷 0 66. 9x 2 16y 2 36x 64y 44 苷 0 67. 25x 2 9y 2 50x 72y 56 苷 0 Let P be a point at which the waves intersect. In the diagram above, 兩F1P F2P兩苷 2.

68. 共x 3兲2 共 y 4兲2 苷共x 1兲2

a. What curve is generated by connecting all points P for which 兩F1P F2P兩苷 2?

Connecting Concepts In Exercises 69 to 72, use the definition of a hyperbola to find the equation of the hyperbola in standard form.

69. Foci 共2, 0兲 and 共2, 0兲; passes through the point 共2, 3兲 70. Foci 共0, 3兲 and 共0, 3兲; passes through the point

71. Foci 共0, 4兲 and 共0, 4兲; passes through the point

冉冊冉冊冉冊 5 ,3 2

73. Sketch a graph of

74. ECCENTRICITY Which of the following hyperbolas has the larger eccentricity? a.

7 ,4 3

72. Foci 共5, 0兲 and 共5, 0兲; passes through the point 5,

y兩 y兩 x兩x兩苷 1. 16 9

−9.4

9 4

b.

6.2

9.4

−6.2

6.2

−9.4

9.4

−6.2

Projects 1. a.

A HYPERBOLIC PARABOLOID A hyperbolic paraboloid is a three-dimensional figure. Make a drawing of a hyperbolic paraboloid. Explain the relationship that exists between the equations of the parabolic cross sections and the relationship that exists between the equations of the hyperbolic cross sections.

b.

A HYPERBOLOID OF ONE SHEET Make a sketch of a hyperboloid of one sheet. Explain the different cross sections of the hyperboloid of one sheet. Do some research on nuclear power plants, and explain why nuclear cooling towers are designed in the shape of hyperboloids of one sheet.

6.4

Section 6.4 ■

■

■

The Rotation Theorem for Conics The Conic Identification Theorem Use a Graphing Utility to Graph Second-Degree Equations in Two Variables

Rotation of Axes

403

Rotation of Axes P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A28.

PS1. Expand cos共a b兲. [3.2] PS2. Expand sin共a b兲. [3.2] PS3. Solve cot 2a 苷

PS4. If sin a 苷

p 兹3 for 0 a . [3.6] 3 2

1 兹3 and cos a 苷 , with 0° a 360°, find a. [3.6] 2 2

PS5. Identify the graph of 4x 2 6y 2 9x 16y 8 苷 0. [6.3] PS6. Graph: 4x y 2 2y 3 苷 0 [6.1]

The Rotation Theorem for Conics The equation of a conic with axes parallel to the coordinate axes can be written in a general form. General Equation of a Conic with Axes Parallel to Coordinate Axes The general equation of a conic with axes parallel to the coordinate axes and not both A and C equal to zero is Ax 2 Cy 2 Dx Ey F 苷 0 The graph of the equation is a parabola when AC 苷 0, an ellipse when AC 0, and a hyperbola when AC 0.

The terms Dx, Ey, and F determine the translation of the conic from the origin. The general equation of a conic is a second-degree equation in two variables. A more general second-degree equation can be written that contains a Bxy term.

take note Some choices of the constants A,

B, C, D, E, and F may result in a degenerate conic or an equation

General Second-Degree Equation in Two Variables The general second-degree equation in two variables is Ax 2 Bxy Cy 2 Dx Ey F 苷 0

that has no solutions.

The Bxy term 共B 苷 0兲 determines a rotation of the conic so that its axes are no longer parallel to the coordinate axes.

404

Chapter 6

Topics in Analytic Geometry

y y'

P(x, y) P'(x', y')

R(0, y)

x'

r

R'(0', y')

Q'(x', 0')

θ α O

Q(x, 0)

Figure 6.52

x

A rotation of axes is a rotation of the x- and y-axes about the origin to another position denoted by x and y . We denote the measure of the angle of rotation by a. Let P be some point in the plane, and let r represent the distance of P from the origin. The coordinates of P relative to the xy-coordinate system and the x y -coordinate system are P共x, y兲 and P共x , y 兲, respectively. Let Q共x, 0兲 and R共0, y兲 be the projections of P onto the x- and the y-axis and let Q 共x , 0 兲 and R 共0 , y 兲 be the projections of P onto the x - and the y -axis. (See Figure 6.52.) The angle between the x -axis and OP is denoted by u. We can express the coordinates of P in each coordinate system in terms of a and u. x 苷 r cos共u a兲 y 苷 r sin共u a兲

x 苷 r cos u y 苷 r sin u

Applying the addition formulas for cos共u a兲 and sin共u a兲, we get x 苷 r cos共u a兲苷 r cos u cos a r sin u sin a y 苷 r sin共u a兲苷 r sin u cos a r cos u sin a Now, substituting x for r cos u and y for r sin u into these equations yields x 苷 x cos a y sin a y 苷 y cos a x sin a

• x r cos , y r sin

This proves the equations labeled (1) of the following theorem. Rotation-of-Axes Formulas Suppose that an xy-coordinate system and an x y -coordinate system have the same origin and that a is the angle between the positive x-axis and the positive x -axis. If the coordinates of a point P are 共x, y兲 in one system and 共x , y 兲 in the rotated system, then

冎

x 苷 x cos a y sin a y 苷 y cos a x sin a

(1)

冎

x 苷 x cos a y sin a y 苷 y cos a x sin a

(2)

The derivations of the formulas for x and y are left as an exercise. As we have noted, the appearance of the Bxy 共B 苷 0兲 term in the general second-degree equation indicates that the graph of the conic has been rotated. The angle through which the axes have been rotated can be determined from the following theorem. Rotation Theorem for Conics Let Ax 2 Bxy Cy 2 Dx Ey F 苷 0, B 苷 0, be the equation of a conic in an xy-coordinate system, and let a be an angle of rotation such that cot 2a 苷

AC , B

0 2a 180

Then the equation of the conic in the rotated coordinate system will be A 共x 兲2 C 共y 兲2 D x E y F 苷 0

(3)

6.4

405

Rotation of Axes

where 0 2a 180 and A 苷 A cos2 a B cos a sin a C sin2 a C 苷 A sin2 a B cos a sin a C cos2 a D 苷 D cos a E sin a E 苷 D sin a E cos a F 苷 F

(4) (5) (6) (7) (8)

Equation (3) has an infinite number of solutions. Any rotation through an AC angle a that is a solution of cot 2a 苷 will eliminate the xy term. However, B the equations of the conic in the rotated coordinate systems may differ, depending on the value of a that is used to produce these transformed equations. To avoid any confusion, we will always choose a to be the acute angle that satisfies Equation (3). For x 2 4xy 9y 2 6x 8y 20 苷 0, what is the value of cot 2a?

QUESTION

EXAMPLE 1

»

Use the Rotation Theorem to Sketch a Conic

Sketch the graph of 7x 2 6 兹3xy 13y 2 16 苷 0.

Solution We are given A 苷 7,

B 苷 6 兹3,

C 苷 13,

D 苷 0,

E 苷 0,

and F 苷 16

The angle of rotation a can be determined by solving cot 2a 苷

A C 7 13 6 1 兹3 苷苷苷苷 B 3 6 兹3 6 兹3 兹3

This gives us 2a 苷 60 , or a 苷 30 . Because a 苷 30 , we have sin a 苷

1 2

and

cos a 苷

兹3 2

We determine the coefficients A , C , D , E , and F by using Equations (4) to (8).

冉冊共兲冉冊冉冊冉冊冉冊共兲冉冊冉冊冉冊

ANSWER

A 苷 7

兹3 2

C 苷 7

1 2

cot 2 苷

2

2

6 兹3

6 兹3

19 苷2 4

兹3 2

兹3 2

1 2

1 2

13

13

1 2

兹3 2

2

2

苷4 苷 16 Continued

䉴

406

Chapter 6

Topics in Analytic Geometry

冉冊冉冊冉冊冉冊

y

D 苷 0

y'

2

x'

2

E 苷 0

2 α = 30°

−2 −2

2

1 2

0

1 2

0

兹3 苷0 2

苷0

F 苷 F 苷 16

x

2

兹3 2

The equation of the conic in the x y -plane is 4共x 兲2 16共 y 兲2 16 苷 0 or −2

−2

共x 兲2 共 y 兲2 2 苷1 22 1 This is the equation of an ellipse that is centered at the origin of an x y -coordinate system. The ellipse has a semimajor axis a 苷 2 and a semiminor axis b 苷 1. See Figure 6.53.

7x 2 6 兹3xy 13y 2 16 苷 0

Figure 6.53

»

Try Exercise 10, page 410

In Example 1, the angle of rotation a was 30°, which is a special angle. In the next example, we demonstrate a technique that is often used when the angle of rotation is not a special angle. EXAMPLE 2

»

Use the Rotation Theorem to Sketch a Conic

Sketch the graph of 32x 2 48xy 18y 2 15x 20y 苷 0.

Solution We are given A 苷 32,

B 苷 48,

C 苷 18,

D 苷 15,

E 苷 20,

and F 苷 0

Therefore, cot 2a 苷

A C 32 18 7 苷苷 B 48 24

Figure 6.54 shows an angle 2a for which cot 2a 苷

y

7 . From Figure 6.54 we 24

7 . The half-angle identities can be used to 25 determine sin a and cos a. conclude that cos 2a 苷

25

24

sin a 苷 2α −7

Figure 6.54

x

冑

1 共7兾25兲 4 苷 2 5

and

cos a 苷

冑

1 共7兾25兲 3 苷 2 5

A calculator can be used to determine that a ⬇ 53.1 . Equations (4) to (8) give us

冉冊冉冊

A 苷 32

3 5

2

C 苷 32

4 5

2

冉冊冉冊冉冊冉冊冉冊冉冊

共48兲

3 5

4 5

18

4 5

2

共48兲

3 5

4 5

18

3 5

2

苷0 苷 50

6.4

冉冊冉冊 3 5

D 苷共15兲

y'

x'

y

E 苷共15兲

2 2 2

−2

x

−2

32x 48xy 18y 15x 20y 苷 0 2

2

Figure 6.55

共20兲

苷 25

3 5

苷0

The equation of the conic in the x y -plane is 50共 y 兲2 25x 苷 0, or

−2

2

4 5

F 苷 F 苷 0

α = 53.1°

F

冉冊冉冊

共20兲

4 5

407

Rotation of Axes

共 y 兲2 苷

1 x 2

This is the equation of a parabola. Because 4p 苷 focus of the parabola is at

»

冉冊

1 1 , we know p 苷 , and the 2 8

1 , 0 on the x -axis. See Figure 6.55. 8

Try Exercise 20, page 410

The Conic Identification Theorem The following theorem provides us with a procedure that can be used to identify the type of conic that will be produced by graphing an equation that is in the form of the general second-degree equation in two variables. Conic Identification Theorem The graph of Ax 2 Bxy Cy 2 Dx Ey F 苷 0 is either a conic or a degenerate conic. If the graph is a conic, then the graph can be identified by its discriminant B2 4AC. The graph is ■

an ellipse or a circle, provided B2 4AC 0.

■

a parabola, provided B2 4AC 苷 0.

■

a hyperbola, provided B2 4AC 0.

EXAMPLE 3

»

Identify Conic Sections

Each of the following equations has a graph that is a nondegenerate conic. Compute B2 4AC to identify the type of conic given by each equation. a.

2x 2 4xy 2y 2 6x 10 苷 0

b.

2xy 11 苷 0

c.

3x 2 5xy 4y 2 8x 10y 6 苷 0

d.

xy 3y 2 2 苷 0 Continued

䉴

408

Chapter 6

Topics in Analytic Geometry Solution a.

Because B2 4AC 苷共4兲2 4共2兲共2兲苷 0, the graph is a parabola.

b.

Because B2 4AC 苷共2兲2 4共0兲共0兲 0, the graph is a hyperbola.

c.

Because B2 4AC 苷 52 4共3兲共4兲 0, the graph is an ellipse or a circle.

d.

Because B2 4AC 苷 12 4共0兲共3兲 0, the graph is a hyperbola.

»

Try Exercise 32, page 410

Use a Graphing Utility to Graph Second-Degree Equations in Two Variables Integrating Technology To graph a general second-degree equation in two variables with a graphing utility, we first must solve the general equation for y. Consider the general second-degree equation in two variables Ax 2 Bxy Cy 2 Dx Ey F 苷 0

(9)

where A, B, C, D, E, and F are real constants, and C 苷 0. To solve Equation (9) for y, we first rewrite the equation as Cy 2 共Bx E兲y 共Ax 2 Dx F兲苷 0

(10)

Applying the quadratic formula to Equation (10) yields

Integrating Technology A TI-83/TI-83 Plus/TI-84 Plus graphing calculator program is available to graph a rotated conic section by entering its coefficients. This program, ROTATE, can be found on our website at college .hmco.com/info/ aufmannCAT

y苷

共Bx E兲兹共Bx E兲2 4C共Ax 2 Dx F兲 2C

(11)

Thus the graph of Equation (9) can be constructed by graphing both y1 苷

共Bx E兲兹共Bx E兲2 4C共Ax 2 Dx F兲 2C

(12)

y2 苷

共Bx E兲兹共Bx E兲2 4C共Ax 2 Dx F兲 2C

(13)

and

on the same grid.

EXAMPLE 4

»

Use a Graphing Utility to Graph a Conic

Use a graphing utility to graph each conic. a.

7x 2 6xy 2.5y 2 14x 4y 9 苷 0

b.

x 2 5xy 3y 2 25x 84y 375 苷 0

c.

3x 2 6xy 3y 2 15x 12y 8 苷 0

6.4

409

Rotation of Axes

Solution Enter y1 (Equation 12) and y2 (Equation 13) into the function editing menu of a graphing utility. Store the following constants in place of the indicated variables.

a.

1 −2

A 苷 7,

6

B 苷 6,

C 苷 2.5,

D 苷 14,

E 苷 4,

and F 苷 9

Graph y1 and y2 on the same screen. The union of the two graphs is an ellipse. See Figure 6.56. Store the following constants in place of the indicated variables.

b.

A 苷 1,

−9

7x 6xy 2.5y 14x 4y 9 苷 0 2

2

B 苷 5,

C 苷 3,

D 苷 25,

E 苷 84,

and F 苷 375

Graph y1 and y2 on the same screen. The union of the two graphs is a hyperbola. See Figure 6.57.

Figure 6.56

Store the following constants in place of the indicated variables.

c.

A 苷 3,

B 苷 6,

C 苷 3,

D 苷 15, E 苷 12,

and F 苷 8

Graph y1 and y2 on the same screen. The union of the two graphs is a parabola. See Figure 6.58. 40

35

− 20

70 −10

40 −10

−50

x 5xy 3y 25x 84y 375 苷 0

3x 6xy 3y 2 15x 12y 8 苷 0

Figure 6.57

Figure 6.58

2

2

2

»

Try Exercise 26, page 410

Topics for Discussion 1. Two students disagree about the graph of x 2 y 2 苷 4. One student states that the graph of the equation is an ellipse. The other student claims that the equation does not have a graph. Which student is correct? Explain. 2. The graph of 4x 2 y 2 苷 0 consists of two intersecting lines. What are the equations of the lines? 3. The graph of xy 苷 12 is a hyperbola. What are the equations of the asymptotes of the hyperbola? 4. Explain why the graph of the ellipse in Figure 6.56 shows a gap at the left side and the right side of the ellipse. 5. Explain why any conic that is given by a general second-degree equation in which D 苷 0 and E 苷 0 will have a quadratic equation in the rotated coordinate system with D 苷 0 and E 苷 0.

410

Chapter 6

Topics in Analytic Geometry

Exercise Set 6.4 In Exercises 1 to 8, find the acute angle of rotation that eliminates the xy term. State approximate solutions to the nearest 0.1.

22. 5x 2 2xy 10y 2 6x 9y 20 苷 0 23. x 2 6xy y 2 2x 5y 4 苷 0

1. xy 苷 3

24. 2x 2 10xy 3y 2 x 8y 7 苷 0

2. 5x 2 3xy 5y 2 1 苷 0

25. 3x 2 6xy 3y 2 10x 8y 2 苷 0

3. 9x 2 24xy 16y 2 320x 240y 苷 0

26. 2x 2 8xy 8y 2 20x 24y 3 苷 0

4. x 2 4xy 4y 2 6x 5 苷 0 5. 5x 6 兹3xy 11y 4x 3y 2 苷 0 2

2

6. 5x 2 4xy 8y 2 6x 3y 12 苷 0 7. 2x 2 xy y 2 4 苷 0 8. 2x 2 兹3xy 3y 2 2x 6y 36 苷 0 In Exercises 9 to 20, find the acute angle of rotation that eliminates the xy term. Then find an equation in x and y coordinates. Graph the equation.

9. xy 苷 4

27. Find the equations of the asymptotes, relative to an xy-coordinate system, for the hyperbola defined by the equation in Exercise 13. Assume that the xy-coordinate system has the same origin as the x y -coordinate system. 28. Find the coordinates of the foci and the equation of the directrix, relative to an xy-coordinate system, for the parabola defined by the equation in Exercise 16. Assume that the xy-coordinate system has the same origin as the x y -coordinate system. 29. Find the coordinates of the foci, relative to an xy-coordinate system, for the ellipse defined by the equation in Exercise 11. Assume that the xy-coordinate system has the same origin as the x y -coordinate system.

10. xy 苷 10 11. 6x 2 6xy 14y 2 45 苷 0 12. 11x 2 10 兹3xy y 2 20 苷 0 13. x 4xy 2y 1 苷 0 2

In Exercises 30 to 40, use the Conic Identification Theorem to identify the graph of each equation as a parabola, an ellipse (or a circle), or a hyperbola.

30. xy 苷 4

2

14. 9x 2 24xy 16y 2 100 苷 0 15. 3x 2 2 兹3xy y 2 2x 2 兹3y 16 苷 0 16. x 2 2xy y 2 2 兹2x 2 兹2y 苷 0 17. 9x 2 24xy 16y 2 40x 30y 100 苷 0 18. 24x 2 16 兹3xy 8y 2 x 兹3y 8 苷 0 19. 6x 2 24xy y 2 12x 26y 11 苷 0 20. x 2 4xy 4y 2 2 兹5x 兹5y 苷 0 In Exercises 21 to 26, use a graphing utility to graph each equation.

21. 6x 2 xy 2y 2 4x 12y 7 苷 0

31. x 2 xy y 2 40 苷 0 32. 11x 2 10 兹3xy y 2 20 苷 0 33. 3x 2 2 兹3xy y 2 3x 2y 20 苷 0 34. 9x 2 24xy 16y 2 8x 12y 20 苷 0 35. 4x 2 4xy y 2 12y 20 苷 0 36. 5x 2 4xy 8y 2 6x 3y 12 苷 0 37. 5x 2 6 兹3xy 11y 2 4x 3y 2 苷 0 38. 6x 2 6xy 14y 2 14x 12y 60 苷 0 39. 6x 2 2兹3xy 5y 2 3x 2y 20 苷 0 40. 5x 2 2兹3xy 3y 2 x y 12 苷 0

6.4

Rotation of Axes

411

Connecting Concepts 41. By using the rotation-of-axes equations, show that for every choice of a, the equation x 2 y 2 苷 r 2 becomes 共x 兲2 共y 兲2 苷 r 2. 42. The vertices of a hyperbola are 共1, 1兲 and 共1, 1兲. The foci are 共兹2, 兹2 兲 and 共兹2, 兹2 兲. Find an equation of the hyperbola. 43. The vertices on the major axis of an ellipse are the points 共2, 4兲 and 共2, 4兲. The foci are the points 共兹2, 2 兹2 兲 and 共兹2, 2 兹2 兲. Find an equation of the ellipse. 44. The vertex of a parabola is the origin, and the focus is the point 共1, 3兲. Find an equation of the parabola. 45. AN INVARIANT THEOREM Let Ax 2 Bxy Cy 2 Dx Ey F 苷 0 be an equation of a conic in an xy-coordinate system. Let the equation of the conic in the rotated x y -coordinate system be

Show that A C 苷 A C 46. AN INVARIANT THEOREM Let Ax 2 Bxy Cy 2 Dx Ey F 苷 0 be an equation of a conic in an xy-coordinate system. Let the equation of the conic in the rotated x y -coordinate system be A 共x 兲2 B x y C 共y 兲2 D x E y F 苷 0 Show that 共B 兲2 4A C 苷 B2 4AC 47. Use the result of Exercise 46 to verify the Conic Identification Theorem, stated on page 407. 48. Derive the equations labeled (2), page 404, of the rotationof-axes formulas.

A 共x 兲2 B x y C 共 y 兲2 D x E y F 苷 0

Projects 1. USE THE INVARIANT THEOREMS The results of Exercises 45 and 46 illustrate that when the rotation theorem is used to transform equations of the form

Use these equations to transform 10x 2 24xy 17y 2 26 苷 0

Ax 2 Bxy Cy 2 F 苷 0

to the form

A 共x 兲2 C 共 y 兲2 F 苷 0

without applying the rotation theorem.

to the form

the following relationships hold: A C 苷 A C and B2 4AC 苷共B 兲2 4A C

A 共x 兲2 C 共 y 兲2 26 苷 0

412

Chapter 6

Topics in Analytic Geometry

Introduction to Polar Coordinates

Section 6.5 ■

■

■

■

The Polar Coordinate System Graphs of Equations in a Polar Coordinate System Transformations Between Rectangular and Polar Coordinates Write Polar Coordinate Equations as Rectangular Equations and Rectangular Coordinate Equations as Polar Equations

P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A29.

PS1. Is sin x an even or odd function? [2.5] PS2. Is cos x an even or odd function? [2.5] PS3. Solve tan a 苷兹3 for 0 a 2p. [3.6] PS4. If sin a 苷 find a. [3.6]

1 兹3 and cos a 苷 , with 0° a 360°, 2 2

y 6

4 A

2

PS5. Write 共r cos u兲2 共r sin u兲2 in simplest form. [3.1] PS6. For the graph at the right, find the coordinates of point A. Round each coordinate to the nearest tenth. [2.2]

5 32° 2

4

6

x

The Polar Coordinate System

P(r, θ ) r

θ O Pole

Polar axis

Figure 6.59

5π 6

π

3π 4

π 2

2π 3

π 3 π 3, 3

) )

π 4

)4, – 76π )

7π 6

2

)2, 54π ) 5π 4 4π 3

4

)5, – π4 ) 3π 2

Figure 6.60

5π 3

7π 4

π 6

0

11π 6

Until now, we have used a rectangular coordinate system to locate a point in the coordinate plane. An alternative method is to use a polar coordinate system, wherein a point is located by giving a distance from a fixed point and an angle from some fixed direction. A polar coordinate system is formed by drawing a horizontal ray. The ray is called the polar axis, and the initial point of the ray is called the pole. A point P共r, u兲 in the plane is located by specifying a distance r from the pole and an angle u measured from the polar axis to the line segment OP. The angle can be measured in degrees or radians. See Figure 6.59. The coordinates of the pole are 共0, u兲, where u is an arbitrary angle. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the axis. Positive values of r are measured along the ray that makes an angle of u from the polar axis. Negative values of r are measured along the ray that makes an angle of u 180 from the polar axis. See Figures 6.60 and 6.61. In a rectangular coordinate system, there is a one-to-one correspondence between the points in the plane and the ordered pairs 共x, y兲. This is not true for a polar coordinate system. For polar coordinates, the relationship is one-to-many. Infinitely many ordered-pair descriptions correspond to each point P共r, u兲 in a polar coordinate system. For example, consider a point whose coordinates are P共3, 45 兲. Because there are 360° in one complete revolution around a circle, the point P also could be written as 共3, 405 兲, as 共3, 765 兲, as 共3, 1125 兲, and generally as 共3, 45 n 360 兲, where n is an integer. It is also possible to describe the point P共3, 45 兲 by 共3, 225 兲, 共3, 135 兲, and 共3, 315 兲, to name just a few options.

6.5

5π 6

π

3π 4

π 2

2π 3 π − 5, – 4

)

) )− 5, 43π )

π 4

The relationship between an ordered pair and a point is not one-to-many. That is, given an ordered pair 共r, u兲, there is exactly one point in the plane that corresponds to that ordered pair. π 6

Graphs of Equations in a Polar Coordinate System 2

)− 3, π6 )

7π 6

π 3

5π 4 4π 3

5π 3

3π 2

0

4

7π 4

11π 6

Figure 6.61

5π 6

3π 4

2π 3

π 2

π

π 3

2

7π 6

5π 4 4π 3

The graph of u 苷 a is a line through the pole at an angle of a from the polar axis. See Figure 6.63a.

π 6

0

4

7π 5π 4 3

3π 2

11π 6

3π 4

The graph of r sin u 苷 a is a horizontal line passing through the point p a, . See Figure 6.63b. 2

冉冊

The graph of r cos u 苷 a is a vertical line passing through the point 共a, 0兲. See Figure 6.63c. π 2

π 2

α

π 3

π 4

2

5π 4 4π 3

3π 2

r苷2

Figure 6.64

4

5π 3

7π 4

(a, 0)

0

π 6

a. u 苷 a

7π 6

0

0

π 2

π

π 2

(a, π2 )

Figure 6.62

2π 3

A polar equation is an equation in r and u. A solution to a polar equation is an ordered pair 共r, u兲 that satisfies the equation. The graph of a polar equation is the set of all points whose ordered pairs are solutions of the equation. p The graph of the polar equation u 苷 is a line. Because u is independent of 6 p r, u is radian from the polar axis for all values of r. The graph is a line that 6 p makes an angle of radian 共30 兲 from the polar axis. See Figure 6.62. 6 Polar Equations of a Line

π 4

p u苷 6

5π 6

413

Introduction to Polar Coordinates

c. r cos u 苷 a

Figure 6.63

0

11π 6

b. r sin u 苷 a

Figure 6.64 is the graph of the polar equation r 苷 2. Because r is independent of u, r is 2 units from the pole for all values of u. The graph is a circle of radius 2 with center at the pole.

The Graph of r a The graph of r 苷 a is a circle with center at the pole and radius a.

414

Chapter 6 π 2

(r, θ ) θ

0

–θ

(r, –θ )

Symmetry with respect to the line u 苷 0

Topics in Analytic Geometry Suppose that whenever the ordered pair 共r, u兲 lies on the graph of a polar equation, 共r, u兲 also lies on the graph. From Figure 6.65, the graph will have symmetry with respect to the line u 苷 0. Thus one test for symmetry is to replace u by u in the polar equation. If the resulting equation is equivalent to the original equation, the graph is symmetric with respect to the line u 苷 0. Table 6.2 shows the types of symmetry and their associated tests. For each type, if the recommended substitution results in an equivalent equation, the graph will have the indicated symmetry. Figure 6.66 illustrates the tests for symmetry p with respect to the line u 苷 and for symmetry with respect to the pole. 2

Figure 6.65 Table 6.2

Tests for Symmetry

Substitution

Symmetry with respect to

u for u

The line u 苷 0

p u for u, r for r

The line u 苷 0

p u for u

The line u 苷

p 2

u for u, r for r

The line u 苷

p 2

r for r

The pole

p u for u

The pole

π 2

(–r, – θ )

π –θ

π 2

π 2

π 2

(r, θ )

(r, θ )

θ +π

(r, θ ) (r, π – θ )

θ

0

θ

–θ

θ

0

(–r, θ )

Symmetry with respect to the line u 苷

p 2

(r, θ ) θ

0

0

(r, θ + π )

Symmetry with respect to the pole

Figure 6.66

The graph of a polar equation may have a type of symmetry even though a test for that symmetry fails. For example, as we will see later, the graph of r 苷 sin 2u is symmetric with respect to the line u 苷 0. However, using the symmetry test of substituting u for u, we have sin 2共u兲苷 sin 2u 苷 r 苷 r Thus this test fails to show symmetry with respect to the line u 苷 0. The symmetry test of substituting p u for u and r for r establishes symmetry with respect to the line u 苷 0.

6.5 EXAMPLE 1

»

415

Introduction to Polar Coordinates

Graph a Polar Equation

Show that the graph of r 苷 4 cos u is symmetric with respect to the line u 苷 0. Graph the equation.

Solution Test for symmetry with respect to the line u 苷 0. Replace u by u. r 苷 4 cos共u兲苷 4 cos u

5π 6

3π 4

2π 3

π 2

π

7π 6

π 3

2

5π 4 4π 3

3π 2

r 苷 4 cos u

Figure 6.67

π 4

4

5π 3

7π 4

• cos( ) cos

Because replacing u by u results in the original equation r 苷 4 cos u, the graph is symmetric with respect to the line u 苷 0. To graph the equation, begin choosing various values of u and finding the corresponding values of r. However, before doing so, consider two further observations that will reduce the number of u-values you must choose. First, because the cosine function is a periodic function with period 2p, it is only necessary to choose u-values between 0 and 2p (0° and 360°). Second, p 3p when u , cos u is negative, which means that any u between 2 2 these values will produce a negative r. Thus the point will be in the first or fourth quadrant. That is, we need consider only angles u in the first or fourth quadrants. However, because the graph is symmetric with respect to p the line u 苷 0, it is only necessary to choose values of u between 0 and . 2

π 6

0

11π 6

0

6

4

3

2

r

4.0

3.5

2.8

2.0

0.0

By symmetry £¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶§¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶•

6

3.5

4

2.8

3

2.0

2

0.0

The graph of r 苷 4 cos u is a circle with center at 共2, 0兲. See Figure 6.67.

»

Try Exercise 14, page 423

Polar Equations of a Circle The graph of the equation r 苷 a is a circle with center at the pole and radius a. See Figure 6.68a. The graph of the equation r 苷 a cos u is a circle that is symmetric with respect to the line u 苷 0. See Figure 6.68b. The graph of r 苷 a sin u is a circle that is symmetric with respect to the line p u苷 . See Figure 6.68c. 2

416

Chapter 6

Topics in Analytic Geometry π 2

π 2

(a, 0)

π 2

(a, π2 )

(a, 0) 0

0

a. r 苷 a

0

b. r 苷 a cos u

c. r 苷 a sin u

Figure 6.68

Just as there are specifically named curves in an xy-coordinate system (such as parabola and ellipse), there are named curves in an ru-coordinate system. Two of the many types are the limaçon and the rose curve.

Polar Equations of a Limaçon The graph of the equation r 苷 a b cos u is a limaçon that is symmetric with respect to the line u 苷 0. The graph of the equation r 苷 a b sin u is a limaçon that is symmetric p with respect to the line u 苷 . 2 In the special case where 兩a兩苷兩b兩, the graph is called a cardioid.

The graph of r 苷 a b cos u is shown in Figure 6.69 for various values of a and b.

π 2

π 2

π 2

0

0

ⱍⱍ

1 b 2

Convex limaçon (no dimple)

Limaçon (with a dimple)

a b

0

ⱍⱍ

ⱍ ⱍ2

π 2

a

a b

苷1

Cardioid (heart-shaped limaçon)

Figure 6.69

0

ⱍⱍ a b

1

Limaçon (with an inner loop)

6.5

EXAMPLE 2

»

Introduction to Polar Coordinates

417

Sketch the Graph of a Limaçon

Sketch the graph of r 苷 2 2 sin u.

Solution

5π 6

3π 4

2π 3

π 2

π 3

π

7π 6

π 4

4

5π 4 4π 3

3π 2

5π 3

7π 4

π 6

0

From the general equation of a limaçon r 苷 a b sin u with 兩a兩苷兩b兩共兩2兩苷兩2兩兲, the graph of r 苷 2 2 sin u is a cardioid that is symmetric with p respect to the line u 苷 . 2 Because we know that the graph is heart-shaped, we can sketch the graph by finding r for a few values of u. When u 苷 0, r 苷 2. When p 3p u苷 , r 苷 0. When u 苷 p, r 苷 2. When u 苷 , r 苷 4. Sketching a heart2 2 shaped curve through the four points 共2, 0兲,

11π 6

冉冊 0,

p , 2

共2, p兲,

and

冉冊 4,

3p 2

produces the cardioid in Figure 6.70.

»

Try Exercise 20, page 424

r 苷 2 2 sin u

Figure 6.70

Example 3 shows how to use a graphing utility to construct a polar graph.

EXAMPLE 3

»

Use a Graphing Utility to Graph a Polar Equation

Use a graphing utility to graph r 苷 3 2 cos u.

Solution

4

−6

6

From the general equation of a limaçon r 苷 a b cos u, with a 苷 3 and b 苷 2, we know that the graph will be a limaçon with a dimple. The graph will be symmetric with respect to the line u 苷 0. Use polar mode with angle measure in radians. Enter the equation r 苷 3 2 cos u in the polar function editing menu. The graph in Figure 6.71 was produced with a TI-84 by using the following window:

−4

r 苷 3 2 cos u

Figure 6.71

u min=0

Xmin=-6

Ymin=-4

u max=2π

Xmax=6

Ymax=4

u step=0.1

Xscl=1

Yscl=1

»

Try Exercise 26, page 424

When using a graphing utility in polar mode, choose the value of ustep carefully. If ustep is set too small, the graphing utility may require an excessively long period of time to complete the graph. If ustep is set too large, the resulting graph may give only a very rough approximation of the actual graph.

418

Chapter 6

Topics in Analytic Geometry Polar Equations of Rose Curves The graphs of the equations r 苷 a cos nu and r 苷 a sin nu are rose curves. When n is an even number, the number of petals is 2n. See Figure 6.72a. When n is an odd number, the number of petals is n. See Figure 6.72b. π 2

π 2

π

π

0

0

3π 2

3π 2

a. r 苷 a cos 4u a. n 苷 4 is even, 2n 苷 8 petals

b. r 苷 a cos 5u b. n 苷 5 is odd, 5 petals

Figure 6.72 QUESTION

How many petals are in the graph of a. r 苷 4 cos 3u? b. r 苷 5 sin 2u?

EXAMPLE 4

»

Sketch the Graph of a Rose Curve

Sketch the graph of r 苷 2 sin 3u.

Solution

5π 6

3π 4

2π 3

π 2

π

7π 6

π 3

2

5π 4 4π 3

3π 2

r 苷 2 sin 3u

Figure 6.73

π 4

4

5π 3

7π 4

π 6

0

From the general equation of a rose curve r 苷 a sin nu, with a 苷 2 and n 苷 3, the graph of r 苷 2 sin 3u is a rose curve that is symmetric with respect to the p line u 苷 . Because n is an odd number 共n 苷 3兲, there will be three petals 2 in the graph. Choose some values for u and find the corresponding values of r. Use symmetry to sketch the graph. See Figure 6.73.

11π 6

0

18

6

5 18

3

7 18

2

r

0.0

1.0

2.0

1.0

0.0

1.0

2.0

»

Try Exercise 16, page 423

ANSWER

a. Because 3 is an odd number, there are three petals in the graph. b. Because 2 is an even number, there are 2共2兲苷 4 petals in the graph.

6.5 EXAMPLE 5

»

Introduction to Polar Coordinates

419

Use a Graphing Utility to Graph a Rose Curve

Use a graphing utility to graph r 苷 4 cos 2u.

Solution

4

−6

6

From the general equation of a rose curve r 苷 a cos nu, with a 苷 4 and n 苷 2, we know that the graph will be a rose curve with 2n 苷 4 petals. The very tip of each petal will be a 苷 4 units away from the pole. Our symmetry tests also indicate that the graph is symmetric with respect to the line u 苷 0, p the line u 苷 , and the pole. 2 Use polar mode with angle measure in radians. Enter the equation r 苷 4 cos 2u in the polar function editing menu. The graph in Figure 6.74 was produced with a TI-84 by using the following window: u min=0 u max=2π u step=0.1

−4

r 苷 4 cos 2u

Figure 6.74

Xmin=-6 Xmax=6 Xscl=1

Ymin=-4 Ymax=4 Yscl=1

»

Try Exercise 32, page 424

Transformations Between Rectangular and Polar Coordinates

5π 6

3π 4

2π 3

y

π 2

π (r, θ ) 3 π 4 (x, y) π 6 y

0

x 苷 cos u r

or x 苷 r cos u

11π 6

y 苷 sin u r

or

θ

π

A transformation between coordinate systems is a set of equations that relate the coordinates of a point in one system with the coordinates of the point in a second system. By superimposing a rectangular coordinate system on a polar system, we can derive the set of transformation equations. Construct a polar coordinate system and a rectangular system such that the pole coincides with the origin and the polar axis coincides with the positive x-axis. Let a point P have coordinates 共x, y兲 in one system and 共r, u兲 in the other 共r 0兲. From the definitions of sin u and cos u, we have

4

x

x 7π 6

5π 4 4π 3

3π 2

Figure 6.75

7π 5π 4 3

y 苷 r sin u

It can be shown that these equations are also true when r 0. Thus, given the point 共r, u兲 in a polar coordinate system (see Figure 6.75), the coordinates of the point in the xy-coordinate system are given by x 苷 r cos u

y 苷 r sin u

420

Chapter 6

Topics in Analytic Geometry

冉冊

For example, to find the point in the xy-coordinate system that corresponds to the 2p 2p point 4, in the ru-coordinate system, substitute 4 for r and for u into the 3 3 equations and simplify.

冉冊

The point 4,

冉冊冉冊

x 苷 4 cos

2p 1 苷4 3 2

苷 2

y 苷 4 sin

2p 兹3 苷4 3 2

苷 2 兹3

2p in the ru-coordinate system is 共 2, 2 兹3 兲 in the xy-coordinate 3

system. To find the polar coordinates of a given point in the xy-coordinate system, use the Pythagorean Theorem and the definition of the tangent function. Let P共x, y兲 be a point in the plane, and let r be the distance from the origin to the point P. Then r 苷兹x 2 y 2. From the definition of the tangent function of an angle in a right triangle, tan u 苷 Thus u is the angle whose tangent is

y x

y . The quadrant for u depends on the sign x

of x and the sign of y. The equations of transformation between a polar and a rectangular coordinate system are summarized as follows:

Transformations Between Polar and Rectangular Coordinates Given the point 共r, u兲 in the polar coordinate system, the transformation equations to change from polar to rectangular coordinates are x 苷 r cos u

y 苷 r sin u

Given the point 共x, y兲 in the rectangular coordinate system, the transformation equations to change from rectangular to polar coordinates are r 苷兹x 2 y 2

tan u 苷

y , x

x苷0

where r 0, 0 u 2p, and u is chosen so that the point lies in the p 3p appropriate quadrant. If x 苷 0, then u 苷 or u 苷 . 2 2

6.5 EXAMPLE 6

»

421

Introduction to Polar Coordinates

Transform from Polar to Rectangular Coordinates

冉冊

Find the rectangular coordinates of the points whose polar coordinates 3p are: a. 6, b. 共4, 30 兲 4

Solution Use the transformation equations x 苷 r cos u and y 苷 r sin u. 3p 苷 3 兹2 4

3p 苷 3 兹2 4 3p The rectangular coordinates of 6, are 共 3 兹2, 3 兹2 兲. 4

a.

x 苷 6 cos

b.

x 苷 4 cos 30 苷 2 兹3 y 苷 4 sin 30 苷 2 The rectangular coordinates of 共4, 30 兲 are 共 2 兹3, 2 兲.

»

y 苷 6 sin

冉冊

Try Exercise 44, page 424

EXAMPLE 7

»

Transform from Rectangular to Polar Coordinates

Find the polar coordinates of the point whose rectangular coordinates are 共 2, 2 兹3 兲.

Solution Use the transformation equations r 苷兹x 2 y 2 and tan u 苷

y . x

r 苷兹共2兲2 共2 兹3 兲2 苷兹4 12 苷兹16 苷 4 tan u 苷

2 兹3 苷兹3 2

From this and the fact that 共 2, 2 兹3 兲 lies in the third quadrant, u 苷

冉冊

The polar coordinates of 共 2, 2 兹3 兲 are 4,

4p . 3

4p . 3

»

Try Exercise 48, page 424

Write Polar Coordinate Equations as Rectangular Equations and Rectangular Coordinate Equations as Polar Equations Using the transformation equations, it is possible to write a polar coordinate equation in rectangular form or a rectangular coordinate equation in polar form.

422

Chapter 6

Topics in Analytic Geometry EXAMPLE 8

»

Write a Polar Coordinate Equation in Rectangular Form

Find a rectangular form of the equation r 2 cos 2u 苷 3.

Solution r 2 cos 2u 苷 3 r 共1 2 sin2 u兲苷 3 r 2 2r 2 sin2 u 苷 3 r 2 2共r sin u兲2 苷 3 x 2 y 2 2y 2 苷 3 x2 y 2 苷 3 2

• cos 2 1 2 sin2

• r 2 x 2 y 2; sin

y r

A rectangular form of r 2 cos 2u 苷 3 is x 2 y 2 苷 3.

»

Try Exercise 58, page 424

Sometimes the procedures of squaring each side of a polar equation or multiplying each side of a polar equation by r can be used to write the polar equation in rectangular form. In Example 9 we multiply each side of a polar equation by r to convert the equation from polar to rectangular form.

\ XAMPLE 9 E

»

Write a Polar Coordinate Equation in Rectangular Form

Find a rectangular form of the equation r 苷 8 cos u.

Solution

take note Squaring each side of the transformation equation r 苷兹x 2y 2 produces r 2 苷 x 2y 2. This equation is often used to write polar coordinate equations in rectangular form.

r 苷 8 cos u r 2 苷 8r cos u x 2 y 2 苷 8x 2 共x 8x兲 y 2 苷 0 共x 2 8x 16兲 y 2 苷 16 共x 4兲2 y 2 苷 42

• Multiply each side by r. • Use the equations r 2 x 2 y 2 and x r cos . • Subtract 8x from each side. • Complete the square in x. • Write in standard form.

A rectangular form of r 苷 8 cos u is 共x 4兲2 y 2 苷 42. The graph of each of these equations is a circle with center (4, 0) and radius 4.

»

Try Exercise 50, page 424

In Example 10, we use the transformation equations x 苷 r cos u and y 苷 r sin u to convert a rectangular coordinate equation to polar form.

6.5 EXAMPLE 10

»

Introduction to Polar Coordinates

423

Write a Rectangular Coordinate Equation in Polar Form

Find a polar form of the equation x 2 y 2 2x 苷 3.

Solution x 2 y 2 2x 苷 3 共r cos u兲2 共r sin u兲2 2r cos u 苷 3 r 2共cos2 u sin2 u兲 2r cos u 苷 3 r 2 2r cos u 苷 3

• Use the transformation equations x r cos and y r sin . • Simplify.

A polar form of x 2 y 2 2x 苷 3 is r 2 2r cos u 苷 3.

»

Try Exercise 70, page 424

Topics for Discussion 1. In what quadrant is the point 共2, 150 兲 located? p is a line, a tutor rewrites the equation in 6 p p the form u 苷 0 r. In this form, regardless of the value of r, u 苷 . Use 6 6 an analogous approach to explain why the graph of r 苷 a is a circle.

2. To explain why the graph of u 苷

3. Two students use a graphing calculator to graph the polar equation r 苷 2 sin u. One graph appears to be a circle, and the other graph appears to be an ellipse. Both graphs are correct. Explain. 4. Is the graph of r 2 苷 6 cos 2u a rose curve with four petals? Explain your answer.

Exercise Set 6.5 In Exercises 1 to 8, plot the point on a polar coordinate system.

1. 共2, 60 兲

2. 共3, 90 兲

4. 共2, 400 兲

5.

7.

冉冊 3,

5p 3

冉冊 2,

p 4

8. 共3, p兲

In Exercises 9 to 24, sketch the graph of each polar equation.

3. 共1, 315 兲

9. r 苷 3

10. r 苷 5

冉冊

11. u 苷 2

12. u 苷

13. r 苷 6 cos u

14. r 苷 4 sin u

15. r 苷 4 cos 2u

16. r 苷 5 cos 3u

6.

4,

7p 6

p 3

424

Chapter 6

Topics in Analytic Geometry

17. r 苷 2 sin 5u

18. r 苷 3 cos 5u

19. r 苷 2 3 sin u

20. r 苷 2 2 cos u

21. r 苷 4 3 sin u

22. r 苷 2 4 sin u

23. r 苷 2关1 1.5 sin 共u兲兴

24. r 苷 4共1 sin u兲

In Exercises 25 to 40, use a graphing utility to graph each equation.

25. r 苷 3 3 cos u

26. r 苷 4 4 sin u

27. r 苷 4 cos 3u

28. r 苷 2 sin 4u

29. r 苷 3 sec u

30. r 苷 4 csc u

31. r 苷 5 csc u

32. r 苷 4 sec u

33. r 苷 4 sin 共3.5u兲

34. r 苷 6 cos 共2.25u兲

35. r 苷 u, 0 u 6p

36. r 苷 u, 0 u 6p

37. r 苷 2u, 0 u 2p

38. r 苷

1 , 0 u 4p u

40. r 苷

4 cos 3u cos 5u cos u

39. r 苷

6 cos 7u 2 cos 3u cos u

In Exercises 41 to 48, transform the given coordinates to the indicated ordered pair. Round approximate angle measures to the nearest tenth of a degree.

41. 共 1, 兹3 兲 to 共r, u兲 43.

45.

冉冊冉冊 3,

2p to 共x, y兲 3

0,

p 2

to 共x, y兲

47. 共3, 4兲 to 共r, u兲

42. 共 2 兹3, 2 兲 to 共r, u兲 44.

46.

冉冊冉冊 2,

3,

p 3

to 共x, y兲

5p to 共x, y兲 6

48. 共12, 5兲 to 共r, u兲

In Exercises 49 to 62, find a rectangular form of each of the equations.

49. r 苷 3 cos u

50. r 苷 2 sin u

51. r 苷 3 sec u

52. r 苷 4 csc u

53. r 苷 4

54. u 苷

p 4

55. u 苷

p 6

56. r cos u 苷 4

57. r 苷 tan u 59. r 苷

58. r 苷 cot u

2 1 cos u

60. r 苷

61. r共sin u 2 cos u兲苷 6

2 1 sin u

62. r共2 cos u sin u兲苷 3

In Exercises 63 to 74, find a polar form of each of the equations.

63. y 苷 2

64. x 苷 4

65. y 苷兹3x

66. y 苷 x 2

67. x 苷 3

68. xy 苷 4

69. x 2 y 2 苷 4

70. 2x 3y 苷 6

71. x 2 苷 8y

72. y 2 苷 4y

73. x 2 y 2 苷 25

74. x 2 4y 2 苷 16

In Exercises 75 to 82, use a graphing utility to graph each equation.

冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊

75. r 苷 3 cos u

p 4

76. r 苷 2 sin u

p 6

77. r 苷 2 sin 2u

p 3

78. r 苷 3 cos 2u

p 4

79. r 苷 2 2 sin u

p 6

80. r 苷 3 2 cos u

p 3

81. r 苷 1 3 cos u

p 3

82. r 苷 2 4 sin u

p 4

6.5

425

Introduction to Polar Coordinates

Connecting Concepts Explain why the graph of r 2 苷 cos2 u and the graph of r 苷 cos u are not the same.

83.

Explain why the graph of r 苷 cos 2u and the graph of r 苷 2 cos2 u 1 are identical.

84.

93.

The graph of r 苷 1.5sin u 2.5 cos 4u sin7

is a butterfly curve similar to the one shown below.

In Exercises 85 to 92, use a graphing utility to graph each equation.

85. r 苷 4 cos 2u (lemniscate) 2

5π 6

86. r 2 苷 2 sin 2u (lemniscate) 87. r 苷 2共1 sec u兲 (conchoid) 88. r 苷 2 cos 2u sec u (strophoid)

3π 4

2π 3

π 2

π

7π 6

89. ru 苷 2 (spiral) 90. r 苷 2 sin u cos2 2u (bifolium) 91. r 苷兩u兩

u 15

π 3

2

5π 4 4π 3

4

5π 3

3π 2

π 4

7π 4

π 6

0

11π 6

Use a graphing utility to graph the butterfly curve for

92. r 苷 ln u

a. 0 u 5p

b. 0 u 20p

For additional information on butterfly curves, read “The Butterfly Curve” by Temple H. Fay, The American Mathematical Monthly, vol. 96, no. 5 (May 1989), p. 442.

Projects 1. A POLAR DISTANCE FORMULA Let P1共r1 , u1 兲 and P2共r2 , u2 兲 be two distinct points in the ru -plane. a. Verify that the distance d between the points is d 苷兹共r1 兲2 共r2 兲2 2r1 r2 cos 共u2 u1 兲 b. Use the above formula to find the distance (to the nearest hundredth) between 共3, 60 兲 and 共5, 170 兲. c.

Does the formula d苷兹共r1 兲2 共r2 兲2 2r1 r2 cos 共u1u2 兲 also produce the correct distance between P1 and P2 ? Explain.

2. ANOTHER POLAR FORM FOR A CIRCLE a. Verify that the graph of the polar equation r 苷 a sin u b cos u is a circle. Assume that a and b are not both 0. b. What are the center (in rectangular coordinates) and the radius of the circle?

426

Chapter 6

Polar Equations of the Conics

Section 6.6 ■

■

■

Topics in Analytic Geometry

Polar Equations of the Conics Graph a Conic Given in Polar Form Write the Polar Equation of a Conic

P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A31.

PS1. Find the eccentricity of the graph of

y2 x2 苷 1. [6.2] 25 16

PS2. What is the equation of the directrix of the graph of y 2 苷 4x? [6.1] PS3. Solve y 苷 2共1 yx兲 for y. [1.1] 3 , what is the smallest positive value of x that must 1 sin x be excluded from the domain of f? [1.3/2.2]

PS4. For the function f 共x兲苷

PS5. Let e be the eccentricity of a hyperbola. Which of the following statements is true: e 苷 0, 0 e 1, e 苷 1, or e 1? [6.3] PS6. Write

4 sec x in terms of cos x. [3.1] 2 sec x 1

Polar Equations of the Conics The definition of a parabola was given in terms of a point (the focus) and a line (the directrix). The definitions of both an ellipse and a hyperbola were given in terms of two points (the foci). It is possible to define each conic in terms of a point and a line.

兰

Calculus Connection

Focus-Directrix Definitions of the Conics

Directrix D

P(r, θ ) r

θ

Let F be a fixed point and D a fixed line in a plane. Consider the set of all d共P, F兲 points P such that 苷 e, where e is a constant. The graph is a d共P, D兲 parabola for e 苷 1, an ellipse for 0 e 1, and a hyperbola for e 1. See Figure 6.76.

F Focus

Figure 6.76

The fixed point is a focus of the conic, and the fixed line is a directrix. The constant e is the eccentricity of the conic. Using this definition, we can derive the polar equations of the conics.

6.6

427

Polar Equations of the Conics

Standard Forms of the Polar Equations of the Conics Let the pole be a focus of a conic section of eccentricity e, with directrix d units from the focus. Then the equation of the conic is given by one of the following: r苷

ed (1) 1 e cos u

r苷

Vertical directrix to the right of the pole

r苷

ed (2) 1 e cos u

Vertical directrix to the left of the pole

ed (3) 1 e sin u

r苷

Horizontal directrix above the pole

ed (4) 1 e sin u

Horizontal directrix below the pole

When the equation involves cos u, the line u 苷 0 is an axis of symmetry. p When the equation involves sin u, the line u 苷 is an axis of symmetry. 2 Graphs of examples are shown in Figure 6.77.

5π 6

2π 3

π 2

π 3 π 6

π

11π 6 4π 3

r苷

3π 2

5π 3

ed (1) 1 e cos u

π 2

π 6

π

7π 6

11π 6 4π 3

r苷

11π 6 4π 3

r苷

3π 2

5π 3

ed (3) 1 e sin u

π 2

π 3 π 6

5π 6

0

F

7π 6

ed (2) 1 e cos u

2π 3 π 6

π

5π 3

3π 2

π 3

5π 6 0

F

π 2

2π 3

π 3

5π 6 0

F 7π 6

2π 3

π

0

F

7π 6

11π 6 4π 3

r苷

3π 2

5π 3

ed (4) 1 e sin u

Figure 6.77

QUESTION

Is the graph of r 苷

4 a parabola, an ellipse, or a 5 3 sin u

hyperbola?

ANSWER

The eccentricity is

3 , which is less than 1. The graph is an ellipse. 5

428

Chapter 6

Topics in Analytic Geometry We will derive Equation (2). Let P共r, u兲 be any point on a conic section. Then, by definition,

Directrix D

P(r, θ )

d共P, F兲苷 e or d共P, F兲苷 e d共P, D兲 d共P, D兲

r

θ

A F

From Figure 6.78, d共P, F兲苷 r and d共P, D兲苷 d共A, Q兲. But note that

Q

d共A, Q兲苷 d共A, F兲 d共F, Q兲苷 d r cos u

Pole

Thus d

r 苷 e共d r cos u兲苷 ed er cos u r er cos u 苷 ed ed r苷 1 e cos u

Figure 6.78

• d(P, F ) e d(P, D) • Subtract er cos from each side. • Solve for r.

The remaining equations can be derived in a similar manner.

Graph a Conic Given in Polar Form EXAMPLE 1

»

Sketch the Graph of a Hyperbola Given in Polar Form

Describe and sketch the graph of r 苷

8 . 2 3 sin u

Solution Write the equation in standard form by dividing the numerator and denominator by 2, the constant term in the denominator.

5π 6

3π 4

π 2

2π 3

(4, π)

π

( 7π 6

π 3

5π 4 4π 3

8 3π , 5 2

r苷 π 4

(4, 0) 4

)

8

)− 8, π2 ) 3π 2

Figure 6.79

5π 3

7π 4

π 6

4 3 1 sin u 2

3 1, the graph is a hyperbola 2 with a focus at the pole. Because the equation contains the expression sin u, p the transverse axis is on the line u 苷 . 2 p 3p To find the vertices, choose equal to and . The corresponding 2 2 8 p 8 3p values of r are 8 and . The vertices are 8, and , . By 5 2 5 2 choosing equal to 0 and , we can determine the points 共4, 0兲 and 共4, p兲 on Because e is the coefficient of sin u and e 苷

0

11π 6

冉冊冉冊

6.6

429

Polar Equations of the Conics

the upper branch of the hyperbola. The lower branch can be determined by symmetry. Plot some points 共r, u兲 for additional values of and corresponding values of r. See Figure 6.79.

»

Try Exercise 2, page 431

EXAMPLE 2

»

Sketch the Graph of an Ellipse Given in Polar Form

Describe and sketch the graph of r 苷

4 . 2 cos u

Solution Write the equation in standard form by dividing the numerator and denominator by 2, which is the constant term in the denominator.

5π 6

π

3π 4

2π 3

(4, π )

7π 6

π 2

π 3

π 4

π

)2, 2 ) 4 ) 3 , 0)

3π 2

Figure 6.80

5π 3

π 6

2 1 1 cos u 2

1 and the graph is an ellipse with a focus at the pole. Because 2 the equation contains the expression cos u, the major axis is on the polar axis. To find the vertices, choose equal to 0 and . The corresponding values 4 4 for r are and 4. The vertices on the major axis are , 0 and 共4, p兲. Plot 3 3 some points 共r, u兲 for additional values of and the corresponding values p 3p of r. Two possible points are 2, and 2, . See the graph of the ellipse 2 2 in Figure 6.80. Thus e 苷

0

)2, 32π ) 5π 4 4π 3

r苷

7π 4

11π 6

冉冊冉冊

冉冊

»

Try Exercise 4, page 431

Write the Polar Equation of a Conic EXAMPLE 3

»

冉冊

Find the Equation of a Conic in Polar Form

Find the polar equation of the parabola, shown in Figure 6.81 on page 430, p with vertex at 2, and focus at the pole. 2 Continued

䉴

430

5π 6

π

Chapter 6

3π 4

2π 3

π 2

π 3

Solution π 4

π

)2, 2 )

(4, π )

7π 6

Topics in Analytic Geometry

5π 4 4π 3

π 6

(4, 0)

3π 2

Figure 6.81

5π 3

7π 4

Because the vertex is on the line u 苷

p and the focus is at the pole, the axis 2

p . Thus the equation of the parabola must 2 involve sin u. The parabola has a horizontal directrix above the pole, so the equation has the form of symmetry is the line u 苷

0

11π 6

r苷

ed 1 e sin u

The distance from the vertex to the focus is 2, so the distance from the focus to the directrix is 4. Because the graph of the equation is a parabola, the eccentricity is 1. The equation is r苷

共1兲共4兲 1 共1兲 sin u

r苷

4 1 sin u

• e 1, d 4

»

Try Exercise 24, page 431

QUESTION

In Example 3, why is there no point on the parabola that corre3p sponds to u 苷 ? 2

Topics for Discussion 1. Is the graph of r 苷

12 a parabola? Explain. 2 cos u

2 sec u is an ellipse except for the fact that it has two 2 sec u 1 holes. Where are the holes located?

2. The graph of r 苷

冉冊

3. Are there two different ellipses that have a focus at the pole and a vertex p ? Explain. at 1, 2 6 have a horizontal axis of symmetry 1 sin u or a vertical axis of symmetry? Explain.

4. Does the parabola given by r 苷

ANSWER

3 4 , sin 苷 1. Thus 1 sin 苷 0, and r 苷 is 2 1 sin undefined. When 苷

6.6

Polar Equations of the Conics

431

Exercise Set 6.6 In Exercises 1 to 14, describe and sketch the graph of each equation.

12 1. r 苷 3 6 cos u

8 2. r 苷 2 4 cos u

3. r 苷

8 4 3 sin u

4. r 苷

6 3 2 cos u

5. r 苷

9 3 3 sin u

6. r 苷

5 2 2 sin u

7. r 苷

10 5 6 cos u

8. r 苷

8 2 4 cos u

4 sec u 9. r 苷 2 sec u 1

冉冊

31. Find the polar equation of the hyperbola with a focus at 3p the pole, vertex at 1, , and eccentricity 2. 2

冉冊

32. Find the polar equation of the ellipse with a focus at the 2 3p pole, vertex at 2, , and eccentricity . 2 3 In Exercises 33 to 40, use a graphing utility to graph each equation. Write a sentence that explains how to obtain the graph from the graph of r as given in the exercise listed to the right of each equation.

33. r 苷

3 sec u 10. r 苷 2 sec u 2

11. r 苷

12 csc u 6 csc u 2

12. r 苷

3 csc u 2 csc u 2

13. r 苷

3 cos u 1

14. r 苷

2 sin u 2

34. r 苷

35. r 苷

36. r 苷

In Exercises 21 to 28, find a polar equation of the conic with focus at the pole and the given eccentricity and directrix.

37. r 苷

23. e 苷 1, r sin u 苷 2

22. e 苷

25. e 苷

2 , r sin u 苷 4 3

26. e 苷

1 , r cos u 苷 2 2

39. r 苷

27. e 苷

3 , r 苷 2 sec u 2

28. e 苷

3 , r 苷 2 csc u 4

40. r 苷

29. Find the polar equation of the parabola with a focus at the pole and vertex 共2, p兲. 30. Find the polar equation of the ellipse with a focus at the 1 pole, vertex at 共4, 0兲, and eccentricity . 2

冉冊

p 2 4 cos u 2

(Compare with Exercise 1.)

(Compare with Exercise 2.)

8 (Compare with Exercise 3.) 4 3 sin共u p兲 6

冉冊 p 3

9

冉冊

3 3 sin u 38. r 苷

p 6

8

3 2 cos u

3 , r sin u 苷 1 2

24. e 苷 1, r cos u 苷 2

冉冊

3 6 cos u

In Exercises 15 to 20, find a rectangular equation for each graph in Exercises 1 to 6.

21. e 苷 2, r cos u 苷 1

12

p 6

5

冉冊

p 2 2 sin u 2

(Compare with Exercise 4.)

(Compare with Exercise 5.)

(Compare with Exercise 6.)

10 (Compare with Exercise 7.) 5 6 cos共u p兲 8

冉冊

2 4 cos u

p 3

(Compare with Exercise 8.)

432

Chapter 6

Topics in Analytic Geometry

Connecting Concepts In Exercises 41 to 46, use a graphing utility to graph each equation.

41. r 苷

3 3 sec u

42. r 苷

5 4 2 csc u

43. r 苷

3 1 2 csc u

44. r 苷

4 1 3 sec u

45. r 苷 4 sin 兹2u, 0 u 12p

46. r 苷 4 cos 兹3u, 0 u 8p ed 47. Let P共r, u兲 satisfy the equation r 苷 . Show that 1 e cos u d共P, F兲苷 e. d共P, D兲 48. Show that the equation of a conic with a focus at the pole ed and directrix r sin u 苷 d is given by r 苷 . 1 e sin u

Projects 1. POLAR EQUATION OF A LINE Verify that the polar equation of a line that is d units from the pole is given by r苷

d cos共u up 兲

r 苷 2a cos共u uc 兲

where up is the angle from the polar axis to a line segment that passes through the pole and is perpendicular to the line.

Section 6.7 ■ ■

■ ■

■

Parametric Equations Eliminate the Parameter of a Pair of Parametric Equations Time as a Parameter The Brachistochrone Problem Parametric Equations and Projectile Motion

2. POLAR EQUATION OF A CIRCLE THAT PASSES THROUGH THE POLE Verify that the polar equation of a circle with center 共a, uc 兲 that passes through the pole is given by

where a is the length of the radius and uc is the angle from the polar axis to the radius that connects the center of the circle with the pole.

Parametric Equations P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A31.

PS1. Complete the square of y 2 3y and write the result as the square of a binomial. [1.1] PS2. If x 苷 2t 1 and y 苷 x 2, write y in terms of t. PS3. Identify the graph of

冉冊冉冊 x2 32

2

y3 22

2

苷 1. [6.2]

PS4. Let x 苷 sin t and y 苷 cos t. What is the value of x 2 y 2? [3.1] PS5. If x 苷 t 1 and y 苷 4 t, what is the ordered pair 共x, y兲 that corresponds to t 苷 5? PS6. What are the domain and range of f 共t兲苷 3 cos 2t? Write the answers using interval notation. [2.5]

6.7

433

Parametric Equations

Parametric Equations The graph of a function is a graph for which no vertical line can intersect the graph more than once. For a graph that is not the graph of a function (an ellipse or a hyperbola, for example), it is frequently useful to describe the graph by parametric equations.

兰

Calculus Connection

Curve and Parametric Equations Let t be a number in an interval I. A curve is a set of ordered pairs 共x, y兲, where x 苷 f共t兲,

y 苷 g共t兲 for t I

The variable t is called a parameter, and the equations x 苷 f共t兲 and y 苷 g共t兲 are parametric equations.

For instance, x 苷 2t 1,

for t 共, 兲

y 苷 4t 1

is an example of a pair of parametric equations. By choosing arbitrary values of t, ordered pairs 共x, y兲 can be created, as shown in the table below. t

x 2t 1

y 4t 1

(x, y)

2

5

7

共5, 7兲

0

1

1

共1, 1兲

1 2

0

3

共0, 3兲

2

3

9

共3, 9兲

y

By plotting the points and drawing a curve through the points, a graph of the parametric equations is produced. See Figure 6.82.

4 (−1, 1)

QUESTION

If x 苷 t 2 1 and y 苷 3 t, what ordered pair corresponds to t 苷 3?

(3, 9)

8

−8

−4

(0, 3)

4 −4

(−5, −7) −8

Figure 6.82

ANSWER

共10, 6兲

8

x

434

Chapter 6

Topics in Analytic Geometry EXAMPLE 1

»

Sketch the Graph of a Curve Given in Parametric Form

Sketch the graph of the curve given by the parametric equations x 苷 t 2 t,

y苷t1

for t R

Solution Begin by making a table of values of t and the corresponding values of x and y. Five values of t were arbitrarily chosen for the table that follows. Many more values might be necessary to determine an accurate graph. y 4

2

2

4

6

x

−2 −4

x 苷 t2 t y苷t1

Figure 6.83

t

x t2 t

yt1

(x, y)

2

2

3

共2, 3兲

1

0

2

共0, 2兲

0

0

1

共0, 1兲

1

2

0

共2, 0兲

2

6

1

共6, 1兲

Graph the ordered pairs 共x, y兲 and then draw a smooth curve through the points. See Figure 6.83.

»

Try Exercise 6, page 440

Eliminate the Parameter of a Pair of Parametric Equations It may not be clear from Example 1 and the corresponding graph that the curve is a parabola. By eliminating the parameter, we can write one equation in x and y that is equivalent to the two parametric equations. To eliminate the parameter, solve y 苷 t 1 for t. y苷t1

or

t苷y1

Substitute y 1 for t in x 苷 t t and then simplify. 2

x 苷共 y 1兲2 共 y 1兲 x 苷 y 2 3y 2

• The equation of a parabola

Complete the square and write the equation in standard form.

冉冊冉冊 x

1 4

苷

y

3 2

2

• This is the equation of a 1 3 parabola vertex at , . 4 2

冉

冊

6.7 EXAMPLE 2

»

435

Parametric Equations

Eliminate the Parameter and Sketch the Graph of a Curve

Eliminate the parameter and sketch the curve of the parametric equations x 苷 sin t,

y 苷 cos t for 0 t 2p

Solution The process of eliminating the parameter sometimes involves trigonometric identities. To eliminate the parameter for the equations, square each side of each equation and then add. x 2 苷 sin2 t y 2 苷 cos2 t x 2 y 2 苷 sin2 t cos2 t y

Thus, using the trigonometric identity sin2 t cos2 t 苷 1, we get

1

x y 苷1 2

2

This is the equation of a circle with center 共0, 0兲 and radius equal to 1. See Figure 6.84.

−1

1

x

−1

Figure 6.84

»

Try Exercise 12, page 440

A parametric representation of a curve is not unique. That is, it is possible that a curve may be given by many different pairs of parametric equations. We will demonstrate this by using the equation of a line and providing two different parametric representations of the line. Consider a line with slope m passing through the point 共x1 , y1 兲. By the pointslope formula, the equation of the line is y y1 苷 m共x x1 兲 Let t 苷 x x1 . Then y y1 苷 mt. A parametric representation is x 苷 x1 t,

y 苷 y1 mt for t a real number

(1)

Let x x1 苷 cot t. Then y y1 苷 m cot t. A parametric representation is x 苷 x1 cot t,

y 苷 y1 m cot t for 0 t p

It can be verified that Equations (1) and (2) represent the original line.

(2)

436

Chapter 6

Topics in Analytic Geometry Example 3 illustrates that the domain of the parameter t can be used to determine the domain and range of the curve defined by the parametric equations. EXAMPLE 3

»

Sketch the Graph of a Curve Given by Parametric Equations

Eliminate the parameter and sketch the graph of the curve that is given by the parametric equations x 苷 2 3 cos t,

y 苷 3 2 sin t for 0 t p

Solution Solve each equation for its trigonometric function. x2 苷 cos t 3

y3 苷 sin t 2

(3)

Using the trigonometric identity cos2 t sin2 t 苷 1, we have cos2 t sin2 t 苷

冉冊冉冊 x2 3

2

y3 2

2

苷1

共x 2兲2 共 y 3兲2 苷1 9 4 This is the equation of an ellipse with center at 共2, 3兲 and major axis parallel to the x-axis. However, because 0 t p, it follows that 1 cos t 1 and 0 sin t 1. Therefore, we have

y

4 (−1, 3)

1

(5, 3) 2

x2 1 3

0

y3 1 2

• Using Equations (3)

Solving these inequalities for x and y yields −2

2

Figure 6.85

4

1 x 5

x

and

3y5

Because the values of y are between 3 and 5, the graph of the parametric equations is only the top half of the ellipse. See Figure 6.85.

»

Try Exercise 14, page 440

Time as a Parameter Parametric equations are often used to show the movement of a point or object as it travels along a curve. For instance, consider the parametric equations x 苷 t 2,

y 苷 t 1 for 2 t 3

For any given value of t in the interval [2, 3], we can evaluate x 苷 t 2 and y 苷 t 1 to determine a point (x, y) on the curve defined by the equations. See the following table and Figure 6.86.

6.7 y 4

t=1 (1, 2)

t=3 (9, 4)

t=2 (4, 3)

t=0 (0, 1)

−2

(1, 0) t = −1

4

x

8

(4, −1) t = −2

Figure 6.86

Parametric Equations

t

x t2

yt1

(x, y)

2

4

1

共4, 1兲

1

1

0

共1, 0兲

0

0

1

共0, 1兲

1

1

2

共1, 2兲

2

4

3

共4, 3兲

3

9

4

(9, 4)

437

As t takes on values from 2 to 3, the curve defined by the parametric equations is traced in a particular direction. The direction in which the curve is traced—by increasing values of the parameter t—is referred to as its orientation. In Figure 6.86 the arrowheads show the orientation of the curve. In Example 4 we let the parameter t represent time. Then the parametric equations x 苷 f(t) and y 苷 g(t) indicate the x- and y-coordinates of a moving point as a function of t. EXAMPLE 4

»

Describe the Motion of a Point

A point moves in a plane such that its position P(x, y) at time t is x 苷 sin t,

y 苷 cos t for 0 t 2p

Describe the motion of the point. y

Solution

(0, 1)

(−1, 0)

(1, 0) x

(0, −1)

Figure 6.87

In Example 2 we determined that the graph of x 苷 sin t, y 苷 cos t for 0 t 2p is a circle with center (0, 0) and radius 1. When t 苷 0, the point P p is at (sin 0, cos 0) 苷 (0, 1). When t 苷 , P is at (1, 0); when t 苷 p, P is at 2 3p (0, 1); when t 苷 , P is at (1, 0); and when t 苷 2p, P is back to its 2 starting position (0, 1). Thus P starts at the point (0, 1) and rotates clockwise around the circle with center (0, 0) and radius 1 as the time t increases from 0 to 2p. See Figure 6.87.

»

Try Exercise 22, page 440

The Brachistochrone Problem

Figure 6.88

One famous problem, involving a bead traveling down a frictionless wire, was posed in 1696 by the mathematician Johann Bernoulli. The problem was to determine the shape of a wire a bead could slide down such that the distance between two points was traveled in the shortest time. Problems that involve “shortest time” are called brachistochrone problems. They are very important in physics and form the basis for much of the classical theory of light propagation. The answer to Bernoulli’s problem is an arc of an inverted cycloid. See Figure 6.88. A cycloid is formed by letting a circle of radius a roll on a straight line without slipping. See Figure 6.89. The curve traced by a point on the circumference of the circle is a cycloid. To find an equation for this curve, begin by placing

438

Chapter 6

Topics in Analytic Geometry a circle tangent to the x-axis with a point P on the circle and at the origin of a rectangular coordinate system. Roll the circle along the x-axis. After the radius of the circle has rotated through an angle u, the coordinates of the point P共x, y兲 can be given by x 苷 h a sin u,

y 苷 k a cos u

(4)

where C共h, k兲 is the current center of the circle. Because the radius of the circle is a, k 苷 a. See Figure 6.89. Because the circle rolls without slipping, the arc length subtended by u equals h. Thus h 苷 au. Substituting for h and k in Equations (4), we have, after factoring, x 苷 a共u sin u兲,

y 苷 a共1 cos u兲 for u 0

See Figure 6.90. y

y

C(h, k) a

P (x, y) k

θ

a cos θ

2a

a sin θ y = k − a cos θ x

x = h − a sin θ

πa

2π a

3π a

h

x 苷 a共u sin u兲, y 苷 a共1 cos u兲

Figure 6.89

Figure 6.90 A cycloid

EXAMPLE 5

»

Graph a Cycloid

Use a graphing utility to graph the cycloid given by x 苷 4共u sin u兲,

y 苷 4共1 cos u兲 for 0 u 4p

Solution Although u is the parameter in the above equations, many graphing utilities, such as the TI-83/TI-83 Plus/TI-84 Plus, use T as the parameter for parametric equations. Thus to graph the equations for 0 u 4p, we use Tmin 苷 0 and Tmax 苷 4p, as shown below. Use radian mode and parametric mode to produce the graph in Figure 6.91.

10

−6

16π

−4

x 苷 4共u sin u兲 y 苷 4共1 cos u兲

Figure 6.91

»

Tmin=0

Xmin=-6

Ymin=-4

Tmax=4π

Xmax=16π

Ymax=10

Tstep=0.5

Xscl=2π

Yscl=1

Try Exercise 30, page 441

x

6.7

Parametric Equations

439

Parametric Equations and Projectile Motion The path of a projectile (assume air resistance is negligible) that is launched at an angle u from the horizon with an initial velocity of v0 feet per second is given by the parametric equations x 苷共v0 cos u兲t,

y 苷 16t 2 共v0 sin u兲t

where t is the time in seconds since the projectile was launched. EXAMPLE 6

»

Graph the Path of a Projectile

Use a graphing utility to graph the path of a projectile that is launched at an angle of u 苷 32 with an initial velocity of 144 feet per second. Use the graph to determine (to the nearest foot) the maximum height of the projectile and the range of the projectile. Assume the ground is level.

Solution Use degree mode and parametric mode. Graph the parametric equations x 苷共144 cos 32 兲t,

y 苷 16t 2 共144 sin 32 兲t for 0 t 5

to produce the graph in Figure 6.92. Use the TRACE feature to determine that the maximum height of 91 feet is attained when t ⬇ 2.38 seconds and that the projectile strikes the ground about 581 feet downrange when t ⬇ 4.76 seconds. 120

Path of projectile

t = 2.38 s (290.64, 90.98)

t = 4.76 s (581.28, 0)

0

600

0

x 苷共144 cos 32 兲t y 苷 16t 2 共144 sin 32 兲t

Figure 6.92

In Figure 6.92, the angle of launch does not appear to be 32° because 1 foot on the x-axis is smaller than 1 foot on the y-axis.

»

Try Exercise 34, page 441

Topics for Discussion 1. It is always possible to eliminate the parameter of a pair of parametric equations. Do you agree? Explain. 2. The line y 苷 3x 5 has more than one parametric representation. Do you agree? Explain.

440

Chapter 6

Topics in Analytic Geometry 3. Parametric equations are used only to graph functions. Do you agree? Explain. 4. Every function y 苷 f共x兲 can be written in parametric form by letting x 苷 t and y 苷 f共t兲. Do you agree? Explain.

Exercise Set 6.7 In Exercises 1 to 10, graph the parametric equations by plotting several points.

1. x 苷 2t, y 苷 t, for t R 2. x 苷 3t, y 苷 6t, for t R

In Exercises 19 to 24, the parameter t represents time and the parametric equations x f (t ) and y g(t ) indicate the x- and y-coordinates of a moving point P as a function of t. Describe the motion of the point as t increases.

19. x 苷 2 3 cos t, y 苷 3 2 sin t, for 0 t p

3. x 苷 t, y 苷 t 1, for t R 2

4. x 苷 2t, y 苷 2t t 1, for t R 2

3p 2

20. x 苷 sin t, y 苷 cos t, for 0 t

5. x 苷 t 2, y 苷 t 3, for t R

21. x 苷 2t 1, y 苷 t 1, for 0 t 3

6. x 苷 t 2 1, y 苷 t 2 1, for t R

22. x 苷 t 1, y 苷兹t, for 0 t 4

7. x 苷 2 cos t, y 苷 3 sin t, for 0 t 2p

23. x 苷 tan

8. x 苷 1 sin t, y 苷 1 cos t, for 0 t 2p 9. x 苷 2 , y 苷 2 , for t R t

冉冊

冉冊

p p p t , y 苷 sec t , for 0 t 4 4 2

24. x 苷 1 t, y 苷 t 2, for 0 t 2

t1

25. Eliminate the parameter for the curves

10. x 苷 t 1, y 苷兹t, for 0 t 9 In Exercises 11 to 18, eliminate the parameter and graph the equation.

11. x 苷 sec t, y 苷 tan t, for

p p t 2 2

and

13. x 苷 2 t 2, y 苷 3 2t 2, for t R

for 0 t

p 16. x 苷 3 sin t, y 苷 cos t, for 0 t 2 17. x 苷兹t 1, y 苷 t, for t 1 18. x 苷兹t, y 苷 2t 1, for t 0

C2 :

x 苷 2 t,

y 苷 1 2t 2 y 苷 1 2t

26. Eliminate the parameter for the curves

and

15. x 苷 cos3 t, y 苷 sin3 t, for 0 t 2p

x 苷 2 t 2,

and then discuss the differences between their graphs.

12. x 苷 3 2 cos t, y 苷 1 3 sin t, for 0 t 2p

14. x 苷 1 t 2, y 苷 2 t 2, for t R

C1 :

C1 :

x 苷 sec2 t,

C2 :

x 苷 1 t 2,

y 苷 tan2 t y 苷 t2

p , and then discuss the differences between 2 their graphs. 27. Sketch the graph of x 苷 sin t,

y 苷 csc t for 0 t

p 2

Sketch another graph for the same pair of equations but 3p choose the domain of t to be p t . 2

6.7 32.

28. Discuss the differences between the graphs of

and

C1 :

x 苷 cos t,

y 苷 cos2 t

C2 :

x 苷 sin t,

y 苷 sin2 t

for 0 t p. 29.

Use a graphing utility to graph the cycloid x 苷 2共t sin t兲, y 苷 2共1 cos t兲 for 0 t 2p.

30.

Use a graphing utility to graph the cycloid x 苷 3共t sin t兲, y 苷 3共1 cos t兲 for 0 t 12p.

441

Parametric Equations

SIMULATE UNIFORM MOTION A Learjet is flying west at 420 miles per hour. A twin engine Piper Seneca is flying north at 235 miles per hour. Both Planes are flying at the same altitude. At time t 苷 0 hours, the Learjet is 800 miles from the intersection point of the flight paths and the Piper Seneca is 415 miles from the intersection point. y 300

In Exercises 31 and 32, the parameter t represents time and the parametric equations x f (t ) and y g (t ) indicate the xand y-coordinates of a moving object as a function of t.

31.

SIMULATE UNIFORM MOTION A Corvette is traveling east at 65 miles per hour. A Hummer is traveling north at 60 miles per hour. Both vehicles are heading toward the same intersection. At time t 苷 0 hours, the Corvette is 6 miles from the intersection and the Hummer is 5 miles from the intersection. y

6

Corvette

2

W

2

4

E S

Hummer 8

10

x

a. The location of the Corvette is described by the equations x 苷 65t, y 苷 5, for t 0. Find a pair of parametric equations that describes the location of the Hummer at the time t, where t is measured in hours. b. Graph the parametric equations of the vehicles to produce a simulation. In the MODE menu, select SIMUL so that the motion of both vehicles will be shown simultaneously. Use the following window settings.

Tmin=0 Xmin=–0.6 Tmax=0.11 Xmax=10 Tstep=0.0005 Xscl=1

−600

−400

100 −200

200

400

x

N (−500, −215)

−200

W

E S

a. The location of the Piper Seneca is described by x 苷 500, y 苷 215 235t, t 0. Find a pair of parametric equations that describes the location of the Learjet at time t, where t is measured in hours.

Tmin=0 TMAX=2 Tstep=0.005

N

5 mi

Intersection point

b. Graph the parametric equations of the planes to produce a simulation. In the MODE menu, select SIMUL so that the motion of both planes will be shown simultaneously. Use the following window settings.

Intersection

6 mi

4

(300, 200)

(−500, 200)

Ymin=0 Ymax=7 Yscl=1

Which vehicle was the first to reach the intersection?

Xmin=–600 Xmax=400 Xscl=100

Ymin=–330 Ymax=330 Yscl=100

Which plane was the first to reach the intersection point? In Exercises 33 to 36, graph the path of the projectile that is launched at an angle of with the horizon with an initial velocity of v0. In each exercise, use the graph to determine the maximum height and the range of the projectile (to the nearest foot). Also state the time t at which the projectile reaches its maximum height and the time it hits the ground. Assume that the ground is level and the only force acting on the projectile is gravity.

33. u 苷 55 , v0 苷 210 feet per second 34. u 苷 35 , v0 苷 195 feet per second 35. u 苷 42 , v0 苷 315 feet per second 36. u 苷 52 , v0 苷 315 feet per second

442

Chapter 6

Topics in Analytic Geometry

Parametric equations of the form x a sin t, y b cos t, for t 0, are encountered in electrical circuit theory. The graphs of these equations are called Lissajous figures. In Exercises 37 to 40, use Tstep 0.05 and 0 t 2 .

38. Graph: x 苷 5 sin 3t, y 苷 5 cos 2t 39. Graph: x 苷 4 sin 2t, y 苷 4 cos 3t 40. Graph: x 苷 5 sin 10t, y 苷 5 cos 9t

37. Graph: x 苷 5 sin 2t, y 苷 5 cos t

Connecting Concepts 41. Let P1共x1 , y1 兲 and P2共x2 , y2 兲 be two distinct points in the plane, and consider the line L passing through those points. Choose a point P共x, y兲 on the line L. Show that x x1 y y1 苷 x 2 x1 y2 y1

44. A circle of radius a rolls without slipping on the outside of a circle of radius b a. Find the parametric equations of a point P on the smaller circle. The curve is called an epicycloid. y

Use this result to demonstrate that x 苷共x2 x1 兲t x1 , y 苷共 y2 y1 兲t y1 is a parametric representation of the line through the two points.

a P(x, y)

42. Show that x 苷 h a sin t, y 苷 k b cos t, for a 0, b 0, and 0 t 2p, are parametric equations for an ellipse with center at 共h, k兲. 43. Suppose a string, held taut, is unwound from the circumference of a circle of radius a. The path traced by the end of the string is called the involute of a circle. Find parametric equations for the involute of a circle.

θ x

A(b, 0)

45. A circle of radius a rolls without slipping on the inside of a circle of radius b a. Find the parametric equations of a point P on the smaller circle. The curve is called a hypocycloid. y

y T a

P(x, y)

θ

a A(a, 0)

x

P(x, y)

θ A(b, 0)

x

Projects 1. PARAMETRIC EQUATIONS IN AN XYZ-COORDINATE SYSTEM a. Graph the three-dimensional curve given by x 苷 3 cos t,

y 苷 3 sin t,

z 苷 0.5t

b. Graph the three-dimensional curve given by x 苷 3 cos t,

y 苷 6 sin t,

z 苷 0.5t

c. What is the main difference between these curves? d. What name is given to curves of this type?

Exploring Concepts with Technology

443

Exploring Concepts with Technology Using a Graphing Calculator to Find the nth Roots of z In Chapter 5 we used De Moivre’s Theorem to find the nth roots of a number. The parametric feature of a graphing calculator can also be used to find and display the nth roots of z 苷 r共cos u i sin u兲. Here is the procedure for a TI-83/TI-83 Plus/TI-84 Plus graphing calculator. Put the calculator in parametric and degree mode. See Figure 6.93. To find the nth roots of z 苷 r共cos i sin 兲, enter in the Y= menu

Normal Sci Eng Float 0123456789 Radian Degree Func Par Pol Seq Connected Dot Sequential Simul FullScreen Split

X1T=r^(1/n)cos(/n+T) and Y1T=r^(1/n)sin(/n+T) In the WINDOW menu, set Tmin=0, Tmax=360, and Tstep=360/n. Set Xmin, Xmax, Ymin, and Ymax to appropriate values that will allow the roots to be seen in the graph window. Press GRAPH to display a polygon. The x- and y-coordinates of each vertex of the polygon represent a root of z in the rectangular form x yi. Here is a specific example that illustrates this procedure.

Figure 6.93

Example Find the fourth roots of z 苷 16i. X1T = 16^(1/4)cos(90/4+T) Y1T = 16^(1/4) s i n (90/4+T) X 2T = Y 2T = WINDOW FORMAT Tm i n =0 X3T = Tm a x =360 Y 3T = Tstep=90 Xm i n =-4 Xmax=4 Xscl=1 Ymin=-3

In trigonometric form, z 苷 16共cos 90 i sin 90 兲. Thus, in this example, r 苷 16, 苷 90 , and n 苷 4. In the Y= menu, enter X1T=16^(1/4)cos(90/4+T) and Y1T=16^(1/4)sin(90/4+T) In the WINDOW menu, set

Figure 6.94

1

4

T=0 X=1.8477591

Y=.76536686 −3i

The vertices of the quadrilateral represent the fourth roots of 16i in the complex plane.

Figure 6.95

Xmin=-4

Ymin=-3

Tmax=360

Xmax=4

Ymax=3

Tstep=360/4

Xscl=1

Yscl=1

See Figure 6.94. Press GRAPH to produce the quadrilateral in Figure 6.95. Use TRACE and the arrow key 䉴 to move to each of the vertices of the quadrilateral. Figure 6.95 shows that one of the roots of z 苷 16i is 1.8477591 0.76536686i. Continue to press the arrow key 䉴 to find the other three roots, which are

3i

−4

Tmin=0

0.7653669 1.8477591i, 1.847759 0.7653669i, and 0.76536686 1.847759i Use a graphing calculator to estimate, in rectangular form, each of the following. 1. The cube roots of 27 2. The fifth roots of 32i 3. The fourth roots of 兹8 兹8i 4. The sixth roots of 64i

444

Chapter 6

Topics in Analytic Geometry

Chapter 6 Summary 6.1 Parabolas • The equations of a parabola with vertex at 共h, k兲 and axis of symmetry parallel to a coordinate axis are given by 共x h兲2 苷 4p共 y k兲; focus 共h, k p兲; directrix y 苷 k p 共 y k兲2 苷 4p共x h兲; focus 共h p, k兲; directrix x 苷 h p

then the graph can be identified by its discriminant B2 4AC. The graph is an ellipse or a circle, provided B2 4AC 0. a parabola, provided B2 4AC 苷 0. a hyperbola, provided B2 4AC 0. • The graph of Ax 2 Bxy Cy 2 Dx Ey F 苷 0 can be constructed by using a graphing utility to graph both

6.2 Ellipses • The equations of an ellipse with center at 共h, k兲 and major axis parallel to a coordinate axis are given by 共x h兲2 共 y k兲2 苷 1; foci 共h c, k兲; vertices 共h a, k兲 a2 b2 共x h兲2 共 y k兲2 苷 1; foci 共h, k c兲; vertices 共h, k a兲 b2 a2

共Bx E兲兹共Bx E兲2 4C共Ax 2 Dx F兲 2C

and y2 苷

共Bx E兲兹共Bx E兲2 4C共Ax 2 Dx F兲 2C

6.5 Introduction to Polar Coordinates

For each equation, a b and c2 苷 a2 b 2. • The eccentricity e of an ellipse is given by e 苷

y1 苷

• A polar coordinate system is formed by drawing a horizontal ray (polar axis). The pole is the origin of a polar coordinate system.

c . a

6.3 Hyperbolas • The equations of a hyperbola with center at 共h, k兲 and transverse axis parallel to a coordinate axis are given by 共x h兲2 共 y k兲2 苷 1; foci 共h c, k兲; vertices 共h a, k兲 a2 b2 共 y k兲2 共x h兲2 苷 1; foci 共h, k c兲; vertices 共h, k a兲 a2 b2

• A point is specified by coordinates 共r, 兲, where r is a directed distance from the pole and is an angle measured from the polar axis. The transformation equations between a polar coordinate system and a rectangular coordinate system are y 苷 r sin

Polar to rectangular:

x 苷 r cos u

Rectangular to polar:

r 苷兹x2 y2

tan u 苷

y ,x苷0 x

• The polar equations of:

For each equation, c2 苷 a2 b2. • The eccentricity e of a hyperbola is given by e 苷

c . a

lines are on page 413. circles are on page 415. limaçons are on page 416. rose curves are on page 418.

6.4 Rotation of Axes • The rotation-of-axes formulas are

再

x 苷 x cos y sin y 苷 y cos x sin

再

x 苷 x cos y sin y 苷 y cos x sin

6.6 Polar Equations of the Conics • The polar equations of the conics are given by

• To eliminate the xy term from the general quadratic equation, rotate the coordinate axes through an angle , where cot 2 苷

AC , B

B 苷 0,

0 2 180

• The graph of Ax 2 Bxy Cy 2 Dx Ey F 苷 0 is either a conic or a degenerate conic. If the graph is a conic,

r苷

ed 1 e cos

or

r苷

ed 1 e sin

where e is the eccentricity and d is the distance of the directrix from the focus. When 0 e 1, the graph is an ellipse. e 苷 1, the graph is a parabola. e 1, the graph is a hyperbola.

Assessing Concepts 6.7 Parametric Equations • Let t be a number in an interval I. A curve is a set of ordered pairs 共x, y兲, where x 苷 f 共t兲,

y 苷 g共t兲 for t I

The variable t is called a parameter, and the pair of equations are parametric equations. • To eliminate the parameter is to find an equation in x and y that has the same graph as the given parametric equations.

• Parametric equations can be used to show the movement of a point as it travels along a curve. • A cycloid is traced by a point on a circle as the circle rolls on a line. Parametric equations of a cycloid are given on page 438. • The path of a projectile (assume air resistance is negligible) that is launched at an angle from the horizon with an initial velocity of v0 feet per second is given by x 苷共v0 cos 兲t,

y 苷 16t 2 共v0 sin 兲t

where t is the time in seconds since the projectile was launched.

Chapter 6 Assessing Concepts In Exercises 1 to 12, match an equation to each description. A letter may be used more than once. Some letters may not be needed.

1. Equation, in rectangular form, of an ellipse with major axis of length 10 and minor axis of length 6 _____

a. 3x2 4y2 苷 12

2. Equation, in rectangular form, of a hyperbola with center (4, 3) ______

b.

3. Equation, in rectangular form, of a parabola with vertex (2, 3) ______ 4. Equation, in rectangular form, of an ellipse with eccen1 tricity ______ 2 5. Equation, in rectangular form, of a hyperbola with eccen兹7 tricity of ______ 2 6. Equation, in rectangular form, of a parabola with focus (3, 2) ______

(x 4)2 (y 3)2 苷1 42 52

c. 3x2 4y2 苷 12 d.

y2 x2 苷1 25 9

e. x 2y2 12y 20 苷 0 f. 8x y2 4y 12 苷 0 g. 3x y2 6y 3 苷 0 h. r 苷

5 3 2 cos u 5

7. Equation, in rectangular form, of a parabola with vertex (2, 3) ______

i. r 苷

6 2 cos u

8. Polar equation of an ellipse with major axis on the polar axis ______

j. r 苷

1 2 2 cos u 3 3

9. Polar equation of a hyperbola with transverse axis on the polar axis ______ 10. Polar equation of a parabola with axis of symmetry on the polar axis ______ 11. Parametric equations of an ellipse with major axis parallel to the x-axis ______ 12. Parametric equations of an ellipse with center (2, 3) _____

445

k. x 苷 2 4 cos t, y 苷 3 2 sin t, for 0 t 2p l. x 苷 1 3 sin t, y 苷 3 8 cos t, for 0 t 2p

446

Chapter 6

Topics in Analytic Geometry

Chapter 6 Review Exercises In Exercises 1 to 12, if the equation is that of an ellipse or a hyperbola, find the center, vertices, and foci. For hyperbolas, find the equations of the asymptotes. If the equation is that of a parabola, find the vertex, the focus, and the equation of the directrix. Graph each equation.

1. x 2 y 2 苷 4 2. y 2 苷 16x

19. Hyperbola with vertices 共 6, 0兲 and asymptotes whose 1 equations are y 苷 x. 9 20. Parabola passing through the points 共1, 0兲, 共2, 1兲, and 共0, 1兲 with axis of symmetry parallel to the y-axis. In Exercises 21 to 24, write the equation without an xy term. Name the graph of the equation.

3. x 2 4y 2 6x 8y 3 苷 0

21. 11x 2 6xy 19y 2 40 苷 0

4. 3x 2 4y 2 12x 24y 36 苷 0 5. 3x 4y 8y 2 苷 0

22. 3x 2 6xy 3y 2 4x 5y 12 苷 0

2

23. x 2 2 兹3xy 3y 2 8 兹3x 8y 32 苷 0

6. 3x 2y 2 4y 7 苷 0

24. xy x y 1 苷 0

7. 9x 2 4y 2 36x 8y 4 苷 0 8. 11x 2 25y 2 44x 50y 256 苷 0 9. 4x 2 9y 2 8x 12y 144 苷 0 10. 9x 16y 36x 16y 104 苷 0 2

In Exercises 25 to 34, graph each polar equation.

25. r 苷 4 cos 3

26. r 苷 1 cos

27. r 苷 2共1 2 sin 兲

28. r 苷 4 sin 4

29. r 苷 5 sin

30. r 苷 3 sec

31. r 苷 4 csc

32. r 苷 4 cos

33. r 苷 3 2 cos

34. r 苷 4 2 sin

2

11. 4x 2 28x 32y 81 苷 0 12. x 6x 9y 27 苷 0 2

In Exercises 13 to 20, find the equation of the conic that satisfies the given conditions.

13. Ellipse with vertices at 共7, 3兲 and 共3, 3兲; length of minor axis is 8. 14. Hyperbola with vertices at 共4, 1兲 and 共2, 1兲 and eccent4 ricity 3 15. Hyperbola with foci 共5, 2兲 and 共1, 2兲; length of transverse axis is 4. 16. Parabola with focus 共2, 3兲 and directrix x 苷 6 17. Parabola with vertex 共0, 2兲 and passing through the point 共3, 4兲 18. Ellipse with eccentricity 共0, 1兲

2 and foci 共4, 1兲 and 3

In Exercises 35 to 38, find a polar form of each equation.

35. y 2 苷 16x

36. x 2 y 2 4x 3y 苷 0

37. 3x 2y 苷 6

38. xy 苷 4

In Exercises 39 to 42, find a rectangular form of each equation.

39. r 苷

4 1 cos

41. r 2 苷 cos 2

40. r 苷 3 cos 4 sin 42. 苷 1

In Exercises 43 to 46, graph the conic given by each polar equation.

43. r 苷

4 3 6 sin

44. r 苷

2 1 cos

45. r 苷

2 2 cos

46. r 苷

6 4 3 sin

Quantitative Reasoning In Exercises 47 to 53, eliminate the parameter and graph the curve given by the parametric equations.

Use a graphing utility to graph the conic given by

55.

x 2 4xy 2y 2 2x 5y 1 苷 0

47. x 苷 4t 2, y 苷 3t 1, for t R

Use a graphing utility to graph

56.

48. x 苷 1 t 2, y 苷 3 2t 2, for t R

r苷

49. x 苷 4 sin t, y 苷 3 cos t, for 0 t 2 50. x 苷 sec t, y 苷 4 tan t, for

51. x 苷

t 2 2

57.

1 2 , y 苷 , for t 0 t t

53. x 苷兹t, y 苷 2t, for t 0 Use a graphing utility to graph the cycloid given by x 苷 3共t sin t兲,

y 苷 3共1 cos t兲

for 0 t 18.

6

冉冊

3 sin

4

PHYSICS The path of a projectile (assume air resistance is negligible) that is launched at an angle from the horizon with an initial velocity of v0 feet per second is given by the parametric equations

52. x 苷 1 cos t, y 苷 2 sin t, for 0 t 2

54.

447

x 苷共v0 cos 兲t,

y 苷 16t 2 共v0 sin 兲t

where t is the time in seconds since the projectile was launched. Use a graphing utility to graph the path of a projectile that is launched at an angle of 33° with an initial velocity of 245 feet per second. Use the graph to determine the maximum height of the projectile to the nearest foot.

»» » » » Quantitative Reasoning: The Mathematics of a Rotary Engine »»» Rotor

During the 1950s, Felix Wankel (1902–1988) developed a rotary engine. The engine has several remarkable properties. • A triangular rotor2 (see Figure 6.96) rotates inside a chamber in a manner such that all three vertices of the rotor maintain constant contact with the chamber. • As the rotor rotates, the centroid (center) of the rotor moves on a circular path.

Chamber

• Every time the rotor makes one complete revolution, the centroid makes three complete revolutions around its circular path. The graph of the parametric equations x 苷 3 cosT 0.5 cos(3T),

Figure 6.96

y 苷 3 sinT 0.5 sin(3T), for 0 T 2p

(1)

where T is the counterclockwise angle of rotation as measured from the positive x-axis, yields the curve shown in Figure 6.97 on page 448. The curve has the same shape as the chamber of a rotary engine. The following TI-83/TI-83 Plus/TI-84 Plus program WANKEL simulates the motion of the rotor in a rotary engine. 2The

actual rotor used in a rotary engine is a modified equilateral triangle with curved sides, as shown in Figure 6.96.

448

Chapter 6

Topics in Analytic Geometry

3.3

−5

PROGRAM:WANKEL ClrDraw:Radian:FnOff -5→Xmin:5→Xmax:1→Xscl -3.3→Ymin:3.3→Ymax:1→Yscl For(T,0,2π/3,π/40) Line(3cos(T)+.5cos(3T), 3sin(T)+.5sin(3T), 3cos(T+2π/3)+.5cos(3T), 3sin(T+2π/3)+.5sin(3T)) Line(3cos(T+2π/3)+.5cos(3T), 3sin(T+2π/3)+.5sin(3T), 3cos(T+4π/3)+.5cos(3T), 3sin(T+4π/3)+.5sin(3T)) Line(3cos(T+4π/3)+.5cos(3T), 3sin(T+4π/3)+.5sin(3T), 3cos(T+2π)+.5cos(3T), 3sin(T+2π)+.5sin(3T)) Pt-On(.5cos(3T),.5sin(3T)) End

5

− 3.3

Figure 6.97

QR1. Run the WANKEL program. How does the path that is traced out by the vertices of the rotor compare with the graph in Figure 6.97?

y 4

2

−4

−2

B

2 A −2

E

Figure 6.98

C x

Felix Wankel designed the shape of the chamber of his rotary engine using trial and error. He did not know that mathematicians had made a study of this figure many years before by graphing the path that is traced out by a point on the radius of a circle that rolls around another circle. For example, Figure 6.98 shows a circle rolling around a larger circle. Point B, the midpoint of radius AC of the small circle, generates curve E as the small circle is rolled around the larger circle. Curve E is called an epitrochoid. The following TI-83/TI-83 Plus/TI-84 Plus program EPIT simulates the process of generating an epitrochoid by rolling a small circle around a larger circle and plotting the trace of the midpoint of a radius of the smaller circle (point B in Figure 6.98). PROGRAM:EPIT Radian:FnOff ClrDraw -1→Xmin:6→Xmax 1→Xscl -.33→Ymin:4.3→Ymax 1→Yscl Circle(0,0,2) For(K,0,π/2,π/30) Circle(3cos(K),3sin(K),1) Circle(3cos(K)+.5cos(3K), 3sin(K)+.5sin(3K),.09) End

QR2. Run the EPIT program. How does the path that is traced out by the midpoint of the radius of the small circle compare with the graph in Figure 6.98, in Quadrant I? QR3. Construct a two-dimensional model of a rotary engine. Demonstrate to your classmates how the rotor revolves inside the epitrochoid chamber. One method for constructing the model is to use thick cardboard from which you cut out an epitrochoid. To generate the epitrochoid, use a computer to print a large graph of the parametric equations given in (1).

Test

449

Cut out an equilateral triangle from the cardboard to form the rotor. What is the length, to the nearest tenth of a unit, of each side of the triangle? (Hint: The length of each side of the triangle equals the distance between 2p the points on the epitrochoid (1) at T 苷 0 and T 苷 .) 3 For additional information on rotary engines and epitrochoids, visit website http://auto.howstuffworks.com/rotary-engine.htm and http://www.math .dartmouth.edu/~dlittle/java/SpiroGraph/.

Chapter 6 Test 1. Find the vertex, focus, and directrix of the parabola given 1 by the equation y 苷 x2. 8 2. Graph:

x2 y2 苷1 16 1

3. Find the vertices and foci of the ellipse given by the equation 25x 2 150x 9y 2 18y 9 苷 0. 4. Find the equation in standard form of the ellipse with center 共0, 3兲, foci 共6, 3兲 and 共6, 3兲, and minor axis of length 6. 5. Graph:

y2 x2 苷1 25 16

12. Graph: r 苷 3共1 sin 兲 13. Graph: r 苷 2 sin 4

冉冊

14. Find the rectangular coordinates of the point whose polar 7 coordinates are 5, . 3 15. Find the rectangular form of r r cos 苷 4. 4 as an equation in rectangular 1 sin coordinates.

16. Write r 苷

17. Eliminate the parameter and graph the curve given by the parametric equations x 苷 t 3, y 苷 2t 2.

6. Find the vertices, foci, and asymptotes of the hyperbola y2 x2 given by the equation 苷 1. 36 64

18. Eliminate the parameter and graph the curve given by the parametric equations x 苷 4 sin , y 苷 cos 2, where 0 2.

7. Graph: 16y 2 32y 4x 2 24x 苷 84

19.

x 苷 2共t sin t兲,

8. For the equation x 2 4xy 5y 2 3x 5y 20 苷 0, determine what acute angle of rotation (to the nearest 0.01°) would eliminate the xy term.

8x 2 5xy 2y 2 10x 5y 4 苷 0 10. P共 1, 兹3 兲 are the coordinates of a point in an xycoordinate system. Find the polar coordinates of P. 11. Graph: r 苷 4 cos

y 苷 2共1 cos t兲

for 0 t 12. 20.

9. Determine whether the graph of the following equation is the graph of a parabola, an ellipse, or a hyperbola.

Use a graphing utility to graph the cycloid given by

The path of a projectile that is launched at an angle of 30° from the horizon with an initial velocity of 128 feet per second is given by x 苷共128 cos 30 兲t,

y 苷 16t 2 共128 sin 30 兲t

where t is the time in seconds after the projectile is launched. Use a graphing utility to determine how far (to the nearest foot) the projectile will travel downrange if the ground is level.

450

Chapter 6

Topics in Analytic Geometry

Cumulative Review Exercises 1. Solve: x2 4x 6 苷 0 2. Is the graph of f 共x兲苷 x3 4x symmetric with respect to the x-axis, the y-axis, or the origin, or does it exhibit none of these symmetries?

4. Convert 240° to radians.

13. Solve: cos1 x 苷 sin1

5. An electric cart has 10-inch-radius wheels. What is the linear speed in miles per hour of this cart when the wheels are rotating at 3 radians per second? Round to the nearest mile per hour. 兹3 3 , t 2, find tan t. 2 2

冊

1 . 5

12 13

14. To find the distance across a ravine, a surveying team locates points A and B on one side of the ravine and point C on the other side of the ravine. The distance between A and B is 155 feet. The measure of angle CAB is 71°, and the measure of angle CBA is 80°. Find the distance across the ravine. Round to the nearest foot. 15. The lengths of the sides of a triangular piece of wood are 2.5 feet, 4 feet, and 3.6 feet. Find the angle between the two longer sides of the triangle. Round to the nearest degree.

7. Find the measure of a for the right triangle shown at the right. Round to the nearest centimeter.

a

43°

8. What are the amplitude and period of y 苷

冉冊

x 1 cos ? 2 3

冉冊

9. What is the period of y 苷 2 tan

x ? 3

sin x 苷 csc x cot x. 1 cos x

16. The magnitude of vector v is 30 and the direction angle is 145°. Write the vector in the form v 苷 ai bj, where a and b are rounded to the nearest tenth. 17. Are the vectors v 苷 3i 2j and w 苷 5i 7j orthogonal?

20 cm

10. Verify the identity

冉

12. Find the exact value of sin cos1

3. Given f 共x兲苷 sin x and g共x兲苷 3x 2, find 共g ⴰ f 兲共x兲.

6. Given sin t 苷

3 5 in Quadrant II and cos b 苷 in 5 13 Quadrant III, find sin共兲.

11. Given sin a 苷

18. Write the complex number z 苷 2 2i兹3 in polar form. 19. Sketch the graph of r 苷 3 sin 2u. 20. Write x in terms of y by eliminating the parameter from the parametric equations x 苷 2t 1, y 苷 4t2 1.

7

Exponential and Logarithmic Functions 7.1

Exponential Functions and Their Applications

7.2

Logarithmic Functions and Their Applications

7.3

Properties of Logarithms and Logarithmic Scales

7.4

Exponential and Logarithmic Equations

7.5

Exponential Growth and Decay

7.6

Modeling Data with Exponential and Logarithmic Functions

Modeling Data with an Exponential Function The average annual salary of a basketball player in the National Basketball Association (NBA) has increased from $12,000 a year in 1957 to over $4.5 million a year in 2004. The following bar graph shows the average annual NBA salary for selected years from 1990 to 2002.

NBA player average annual salary (in millions of dollars)

S(t) 4.0 3.5 3.0 2.5

S

2.0 1.5 1.0 0.5 0.0 1990

1994

1998

2002

t

Year

Source: www.sportsfansofamerica.com and the NBA Players Association

SSG

Online Study Center For online student resources, such as section quizzes, visit this website: college.hmco.com/info/aufmannCAT

The function S共t兲苷 0.7739共1.1485兲t closely models the average annual salary of an NBA player for the years 1990 共t 苷 0兲 to 2002 共t 苷 12兲. This function was determined by using the data from the bar graph and exponential regression, which is one of the topics of Section 7.6. See Exercise 23, page 531 for another application in which an exponential function is used to model data.

451

452

Chapter 7

Exponential and Logarithmic Functions

Exponential Functions and Their Applications

Section 7.1 ■ ■

■

Exponential Functions Graphs of Exponential Functions The Natural Exponential Function

Exponential Functions In 1965, Gordon Moore, one of the cofounders of Intel Corporation, observed that the maximum number of transistors that could be placed on a microprocessor seemed to be doubling every 18 to 24 months. This observation is known as Moore’s Law. Table 7.1 below shows how the maximum number of transistors on various Intel processors has changed over time. (Source: Intel Museum home page.) The curve that approximately passes through the points is a mathematical model of the data. See Figure 7.1. The model is based on an exponential function.

Table 7.1 Year Number of transistors per microprocessor (in thousands)

1971

1979

1983

1985

1990

1993

1995

1998

2000

2004

2.3

31

110

280

1200

3100

5500

14,000

42,000

592,000

When light enters water, the intensity of the light decreases with the depth of the water. The graph in Figure 7.2 shows a model, for Lake Michigan, of the decrease in the percentage of available light as the depth of the water increases.

10,000 P 5000

′72 ′76 ′80 ′84 ′88 ′92 ′96 ′00 Year

Figure 7.1 Moore’s Law

Percent of light available

Number of transistors (in thousands)

15,000

80 60 40 20

2

4

Depth (in feet)

Figure 7.2

This model is also based on an exponential function.

Definition of an Exponential Function The exponential function with base b is defined by f共x兲苷 b x where b 0, b 苷 1, and x is a real number.

d

7.1

453

Exponential Functions and Their Applications

The base b of f共x兲苷 b x is required to be positive. If the base were a negative number, the value of the function would be a complex number for some values of x. 1 1 苷共4兲1/2 苷 2i. To avoid complex For instance, if b 苷 4 and x 苷 , then f 2 2 number values of a function, the base of any exponential function must be a positive number. Also, b is defined such that b 苷 1 because f共x兲苷 1x 苷 1 is a constant function. In the following examples we evaluate f共x兲苷 2x at x 苷 3 and x 苷 2.

冉冊

f共3兲苷 23 苷 8

f共2兲苷 22 苷

1 1 2 苷 2 4

To evaluate the exponential function f共x兲苷 2x at an irrational number such as x 苷兹2, we use a rational approximation of 兹2, such as 1.4142, and a calculator to obtain an approximation of the function. For instance, if f共x兲苷 2x, then f 共兹2 兲苷 2兹2 ⬇ 21.4142 ⬇ 2.6651.

EXAMPLE 1

»

Evaluate an Exponential Function

Evaluate f共x兲苷 3x at x 苷 2, x 苷 4, and x 苷 p.

Solution f共2兲苷 32 苷 9 1 1 苷 34 81 f共p兲苷 3p ⬇ 33.1415927 ⬇ 31.54428

f共4兲苷 34 苷

• Evaluate with the aid of a calculator.

»

Try Exercise 2, page 461

Graphs of Exponential Functions The graph of f共x兲苷 2x is shown in Figure 7.3. The coordinates of some of the points on the curve are given in Table 7.2.

Table 7.2 y

f(x) = 2x

x

8

y f (x) 2x

2

f 共2兲苷 22 苷

1 4

1

f 共1兲苷 21 苷

1 2

6 The graph approaches 4 the negative x-axis, but it does not intersect the axis. 2

− 4 −3 −2 −1

Figure 7.3

1

2

3

4

x

(x, y)

冉冊冉冊 2,

1 4

1,

1 2

0

f 共0兲苷 20 苷 1

共0, 1兲

1

f 共1兲苷 2 苷 2

共1, 2兲

2

f 共2兲苷 2 苷 4

共2, 4兲

3

f 共3兲苷 2 苷 8

共3, 8兲

1 2 3

454

Chapter 7

Exponential and Logarithmic Functions Note the following properties of the graph of the exponential function f共x兲苷 2x. ■

The y-intercept is 共0, 1兲.

■

The graph passes through 共1, 2兲.

■

As x decreases without bound (that is, as x l ), f共x兲 l 0.

■

The graph is a smooth, continuous increasing curve.

Now consider the graph of an exponential function for which the base is 1 x between 0 and 1. The graph of f共x兲苷 is shown in Figure 7.4. The coordi2 nates of some of the points on the curve are given in Table 7.3.

冉冊

Table 7.3 x

y

f 共3兲苷

1 2

3

2

f 共2兲苷

1 2

2

1

f 共1兲苷

1 2

1

6 The graph approaches the positive x-axis, but it does not intersect the axis.

f(x) =

1 x 2

()

2

− 4 −3 −2 −1

1

2

3 4

x

0

f 共0兲苷

1

f 共1兲苷

2

f 共2兲苷

Figure 7.4

Note the following properties of the graph of f共x兲苷

1 2

0

1 2

1

1 2

2

■

The y-intercept is 共0, 1兲.

■

The graph passes through 1,

■

As x increases without bound (that is, as x l

■

The graph is a smooth, continuous decreasing curve.

(x, y)

苷8

共3, 8兲

苷4

共2, 4兲

苷2

共1, 2兲

苷1

冉冊 1 2

x

1 2

3

8

4

冉冊冉冊冉冊冉冊冉冊冉冊冉冊

y f (x)

苷

1 2

苷

1 4

x

.

冉冊

1 . 2

), f共x兲 l 0.

共0, 1兲

冉冊冉冊 1,

1 2

2,

1 4

7.1

455

Exponential Functions and Their Applications

The basic properties of exponential functions are provided in the following summary. Properties of f共x兲 bx For positive real numbers b, b 苷 1, the exponential function defined by f共x兲苷 b x has the following properties: 1. The function f is a one-to-one function. It has the set of real numbers as its domain and the set of positive real numbers as its range. 2. The graph of f is a smooth, continuous curve with a y-intercept of 共0, 1兲, and the graph passes through 共1, b兲. 3. If b 1, f is an increasing function and the graph of f is asymptotic to the negative x-axis. [As x l , f共x兲 l , and as x l , f共x兲 l 0. ] See Figure 7.5a. 4. If 0 b 1, f is a decreasing function and the graph of f is asymptotic to the positive x-axis. [As x l , f共x兲 l , and as x l , f共x兲 l 0.] See Figure 7.5b.

y

y

(−2, b ) −2

(2, b ) 2

(1, b ) 1

(−2, b ) (−1, b ) (−3, b ) −2

(0, 1)

−2

a. f (x) =

−1

(1, b ) (2, b ) 1

−1

−3

b x,

(−1, b ) (0, 1)

2

b>1

x

2

−2

b. f (x) =

3

x

2

b x,

(3, b )

0
Figure 7.5

QUESTION

What is the x-intercept of the graph of f共x兲苷

EXAMPLE 2

Graph: g共x兲苷

»

冉冊 1 3

x

?

Graph an Exponential Function

冉冊 3 4

x

Continued ANSWER

䉴

The graph does not have an x-intercept. As x increases without bound, the graph approaches the x-axis, but it does not intersect the x-axis.

456

Chapter 7

Exponential and Logarithmic Functions Solution 3 is less than 1, we know that the graph of g is a 4 decreasing function that is asymptotic to the positive x-axis. The y-intercept 3 of the graph is the point 共0, 1兲, and the graph also passes through 1, . 4 Plot a few additional points (see Table 7.4), and then draw a smooth curve through the points as in Figure 7.6. Because the base

冉冊

Table 7.4 x 3 2 1 2 3

y g (x)

冉冊冉冊冉冊冉冊冉冊 3 4

3

3 4

2

3 4

1

3 4

2

3 4

3

冉冊 3 4

苷

64 27

苷

16 9

苷

苷

x

4 3

9 16

27 苷 64

(x, y)

冉冉冉冉冉

冊冊冊

3,

64 27

2,

16 9

1, 2,

4 3

冊冊

9 16

27 3, 64

y 4 g(x) = 2 −4

−2

() 3 4

2

x

4

x

−2 −4

Figure 7.6

»

Try Exercise 22, page 462

Consider the functions F共x兲苷 2x 3 and G共x兲苷 2x3. You can construct the graphs of these functions by plotting points; however, it is easier to construct their graphs by using translations of the graph of f共x兲苷 2x, as shown in Example 3.

EXAMPLE 3 a.

»

Use a Translation to Produce a Graph

Explain how to use the graph of f共x兲苷 2x to produce the graph of

F共x兲苷 2x 3. b.

Explain how to use the graph of f共x兲苷 2x to produce the graph of

G共x兲苷 2x3.

Solution a.

F共x兲苷 2x 3 苷 f共x兲 3. The graph of F is a vertical translation of f down 3 units, as shown in Figure 7.7.

b.

G共x兲苷 2x3 苷 f共x 3兲. The graph of G is a horizontal translation of f to the right 3 units, as shown in Figure 7.8.

7.1

457

Exponential Functions and Their Applications

f (x) = 2 x

y

y

6

6

4

4 f(x) = 2 x

2

2 G(x) = 2 x − 3

−4

−2

2

4

x

6

−4

−2

2

4

6

x

F(x) = 2 x − 3

Figure 7.7

Figure 7.8

»

Try Exercise 28, page 462

The graphs of some functions can be constructed by stretching, compressing, or reflecting the graph of an exponential function. EXAMPLE 4

»

Use Stretching or Reflecting Procedures to Produce a Graph

Explain how to use the graph of f共x兲苷 2x to produce the graph of

a.

M共x兲苷 2共2x 兲.

Explain how to use the graph of f共x兲苷 2x to produce the graph of

b.

N共x兲苷 2x.

Solution a.

M共x兲苷 2共2x 兲苷 2f共x兲. The graph of M is a vertical stretching of f away from the x-axis by a factor of 2, as shown in Figure 7.9. (Note: If 共x, y兲 is a point on the graph of f 共x兲苷 2x, then 共x, 2y兲 is a point on the graph of M.)

b.

N共x兲苷 2x 苷 f共x兲. The graph of N is the graph of f reflected across the y-axis, as shown in Figure 7.10. (Note: If 共x, y兲 is a point on the graph of f 共x兲苷 2x, then 共x, y兲 is a point on the graph of N.) y

y

8

8

6 f(x) = 2 M(x) = 2(2 x)

−4

−2

6

−x

4

2

2 2

Try Exercise 30, page 462

4

x

f(x) = 2 x

N(x) = 2

4

Figure 7.9

»

x

−4

−2

2

Figure 7.10

4

x

458

Chapter 7

Exponential and Logarithmic Functions

The Natural Exponential Function

Math Matters

The irrational number p is often used in applications that involve circles. Another irrational number, denoted by the letter e, is useful in many applications that involve growth or decay. Definition of e

Leonhard Euler (1707–1783)

Some mathematicians consider Euler to be the greatest mathematician of all time. He certainly was the most prolific writer of mathematics of all time. He made substantial contributions in the areas of number theory, geometry, calculus, differential equations, differential geometry, topology, complex variables, and analysis, to name but a few. Euler was the first to introduce many of the mathematical notations that we use today. For instance, he introduced the symbol i for the square root of 1, the symbol for pi, the functional notation f 共x兲, and the letter e for the base of the natural exponential function. Euler’s computational skills were truly amazing. The mathematician Franc᝺ois Arago remarked, “Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.”

The number e is defined as the number that 1 n 1 n

冉冊

approaches as n increases without bound.

The letter e was chosen in honor of the Swiss mathematician Leonhard Euler. He was able to compute the value of e to several decimal places by evaluating 1 n for large values of n, as shown in Table 7.5. 1 n

冉冊

Table 7.5 Value of n

Value of

1

冉冊 1

1 n

n

2

10

2.59374246

100

2.704813829

1000

2.716923932

10,000

2.718145927

100,000

2.718268237

1,000,000

2.718280469

10,000,000

2.718281693

The value of e accurate to eight decimal places is 2.71828183. Definition of the Natural Exponential Function

兰

Calculus Connection

For all real numbers x, the function defined by f共x兲苷 e x is called the natural exponential function.

A calculator can be used to evaluate e x for specific values of x. For instance, e 2 ⬇ 7.389056, e 3.5 ⬇ 33.115452, and e1.4 ⬇ 0.246597 On a TI-83/TI-83 Plus/TI-84 Plus calculator the e x function is located above the LN key.

7.1

Integrating Technology

To graph f共x兲苷 e x, use a calculator to find the range values for a few domain values. The range values in Table 7.6 have been rounded to the nearest tenth.

The graph of f (x) 苷 ex below was produced on a TI-83/TI-83 Plus/ TI-84 Plus graphing calculator by entering e x in the Y苷 menu. Plot1 Plot2 Plot3 \Y 1 = e^ ( X) \Y 2 = 6 \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 = −4.7

f (x) = e x 4.7

459

Exponential Functions and Their Applications

Table 7.6 x

f(x) e

x

2

1

0

1

2

0.1

0.4

1.0

2.7

7.4

Plot the points given in Table 7.6, and then connect the points with a smooth curve. Because e 1, we know that the graph is an increasing function. To the far left, the graph will approach the x-axis. The y-intercept is 共0, 1兲. See Figure 7.11. Note in Figure 7.12 how the graph of f共x兲苷 e x compares with the graphs of g共x兲苷 2x and h共x兲苷 3x. You may have anticipated that the graph of f共x兲苷 e x would lie between the two other graphs because e is between 2 and 3.

−1

y

h(x) = 3 x y

g(x) = 2 x

f(x) = e x

8

3

6

f (x) = e x

4 1 2

−4

−3

−2 −1

1

2

3

4

x

−1

Figure 7.11

1

2

x

Figure 7.12

Many applications can be modeled effectively by functions that involve an exponential function. For instance, in Example 5 we make use of a function that involves an exponential function to model the temperature of a cup of coffee.

EXAMPLE 5

»

Use a Mathematical Model

A cup of coffee is heated to 160°F and placed in a room that maintains a temperature of 70°F. The temperature T of the coffee, in degrees Fahrenheit, after t minutes is given by T 苷 70 90e0.0485t a.

Find the temperature of the coffee, to the nearest degree, 20 minutes after it is placed in the room.

b.

Use a graphing utility to determine when the temperature of the coffee will reach 90°F. Continued

䉴

460

Chapter 7

take note

Exponential and Logarithmic Functions Solution

methods of solving this type of

T 苷 70 90e0.0485t 苷 70 90e0.0485共20兲 ⬇ 70 34.1 ⬇ 104.1

equation without the use of a

After 20 minutes the temperature of the coffee is about 104°F.

In Example 5b, we use a graphing

a.

utility to solve the equation 0.0485t

90 苷 70 90e

. Analytic

graphing utility will be developed in Section 7.4.

• Substitute 20 for t.

Graph T 苷 70 90e0.0485t and T 苷 90. See the following figure.

b.

170

0

45

Intersection X=31.011905 Y=90

−40

Xscl = 5

Yscl = 20

The graphs intersect at about 共31.01, 90兲. It takes the coffee about 31 minutes to cool to 90°F.

»

Try Exercise 48, page 463

EXAMPLE 6

»

Use a Mathematical Model

The weekly revenue R, in dollars, from the sale of a product varies with time according to the function R共x兲苷

1760 8 14e0.03x

where x is the number of weeks that have passed since the product was put on the market. What will the weekly revenue approach as time goes by?

Solution Method 1 Use a graphing utility to graph R共x兲, and use the TRACE feature to see what happens to the revenue as the time increases. The graph on the right shows that as the weeks go by, the weekly revenue will increase and approach $220.00 per week.

280

Y1=1760/(8+14e^(-.03X))

0

X=400

−40

Xscl = 100

Y=219.99763 Yscl = 100

400

7.1 Method 2

Exponential Functions and Their Applications

461

Write the revenue function in the following form. R共x兲苷

1760 14 8 0.03x e

• 14e0.03x

14 e0.03x

As x increases without bound, e0.03x increases without bound, and the 14 1760 fraction 0.03x approaches 0. Therefore, as x l , R共x兲 l 苷 220. Both e 80 methods indicate that as the number of weeks increases, the revenue approaches $220 per week.

»

Try Exercise 54, page 464

Topics for Discussion 1. Explain how to use the graph of f共x兲苷 2x to produce the graph of g共x兲苷 2共x3兲 4. 2

2. At what point does the function g共x兲苷 ex /2 take on its maximum value? 3. Without using a graphing utility, determine whether the revenue function R共t兲苷 10 e0.05t is an increasing function or a decreasing function. 4. Discuss the properties of the graph of f共x兲苷 b x when b 1. 5. What is the base of the natural exponential function? How is it calculated? What is its approximate value?

Exercise Set 7.1 In Exercises 1 to 8, evaluate the exponential function for the given x -values.

7. j共x兲苷

1. f 共x兲苷 3 ; x 苷 0 and x 苷 4 x

2. f 共x兲苷 5 ; x 苷 3 and x 苷 2 x

3. g共x兲苷 10 x; x 苷 2 and x 苷 3 4. g共x兲苷 4 ; x 苷 0 and x 苷 1 x

5. h共x兲苷

6. h共x兲苷

冉冊冉冊 3 2

x

2 5

x

; x 苷 2 and x 苷 3

; x 苷 1 and x 苷 3

8. j共x兲苷

冉冊冉冊 1 2

x

1 4

x

; x 苷 2 and x 苷 4

; x 苷 1 and x 苷 5

In Exercises 9 to 14, use a calculator to evaluate the exponential function for the given x -value. Round to the nearest hundredth.

9. f 共x兲苷 2x, x 苷 3.2

10. f 共x兲苷 3x, x 苷 1.5

11. g共x兲苷 ex, x 苷 2.2

12. g共x兲苷 ex, x 苷 1.3

13. h共x兲苷 5x, x 苷兹2

14. h共x兲苷 0.5x, x 苷

462

Chapter 7

Exponential and Logarithmic Functions

15. Examine the following four functions and the graphs labeled a, b, c, and d. For each graph, determine which function has been graphed.

a.

f 共x兲苷 5x

g共x兲苷 1 5x

h共x兲苷 5x3

k共x兲苷 5x 3

b.

y

8

4

4

4 x

−4

c.

y

8

d.

4 x y

28. f 共x兲苷 6x, F共x兲苷 6x5

4 x

−4

冉冊冉冊 1 4

x

1 4

x2

g共x兲苷

冉冊冉冊

k共x兲苷 3

b.

29. f 共x兲苷

30. f 共x兲苷

31. f 共x兲苷

x

1 4

1 4

x

32. f 共x兲苷

冉冊冉冊冉冊冉冊 3 2

x

5 2

x

1 3

x

2 3

x

, F共x兲苷

3 2

, F共x兲苷

, F共x兲苷 2

, F共x兲苷

1 2

4

4

35. f(x) 苷 2x, F(x) 苷 (2x4) 4 x

−4

d.

x

x

x

5 2

1 3

2 3

x

x

8

4

4

36. f(x) 苷 2x, F(x) 苷 (2x ) 37. f(x) 苷 0.5x, F(x) 苷 3 0.5x

y

8

4 x

2 3

y

34. f 共x兲苷 ex, F共x兲苷 ex3 1

y

x

33. f 共x兲苷 ex, F共x兲苷 ex 2

8

4 x

5 2

冉冊冋冉冊册冋冉冊册冋冉冊册

8

−4

冉冊冉冊

26. f 共x兲苷 4x, F共x兲苷 4x 3

4

y

38. f(x) 苷 0.5x, F(x) 苷 3(0.5x2 ) 1

4 x

−4

In Exercises 17 to 24, sketch the graph of each function.

19. f 共x兲苷 10

24. f 共x兲苷

4

h共x兲苷

In Exercises 39 to 46, use a graphing utility to graph each function. If the function has a horizontal asymptote, state the equation of the horizontal asymptote.

39. f 共x兲苷

3x 3x 2

40. f 共x兲苷 4 3x

41. f 共x兲苷

ex ex 2

42. f 共x兲苷

18. f 共x兲苷 4x x

x

27. f 共x兲苷 10 x, F共x兲苷 10 x2

f 共x兲苷

17. f 共x兲苷 3x

1 3

22. f 共x兲苷

25. f 共x兲苷 3x, F共x兲苷 3x 2

16. Examine the following four functions and the graphs labeled a, b, c, and d. For each graph, determine which function has been graphed.

−4

x

8

−4

c.

3 2

8

4 x

a.

23. f 共x兲苷

冉冊冉冊

In Exercises 25 to 38, explain how to use the graph of the first function f to produce the graph of the second function F.

−4

y

21. f 共x兲苷

20. f 共x兲苷 6

x

2

ex ex 2

7.1 43. f 共x兲苷 e共x4兲 45. f 共x兲苷 x0 47.

Exponential Functions and Their Applications

44. f 共x兲苷 0.5ex

10 , 1 0.4e0.5x

46. f 共x兲苷 x0

a. What was the monthly income after the 10th month and after the 100th month?

10 , 1 1.5e0.5x

INTERNET CONNECTIONS Data from Forrester Research suggest that the number of broadband [cable and digital subscriber line (DSL)] connections to the Internet can be modeled by f 共x兲苷 1.353共1.9025兲x, where x is the number of years after January 1, 1998, and f 共x兲 is the number of connections in millions.

b. What will the monthly income from the product approach as the time increases without bound? 51.

a. How many broadband Internet connections, to the nearest million, does this model predict will exist on January 1, 2007? b. According to the model, in what year will the number of broadband connections first reach 1 billion? [Hint: Use the intersect feature of a graphing utility to determine the x-coordinate of the point of intersection of the graphs of f 共x兲 and y 苷 1000.] 48.

MEDICATION IN BLOODSTREAM The function A共t兲苷 200e0.014t gives the amount of medication, in milligrams, in a patient’s bloodstream t minutes after the medication has been injected into the patient’s bloodstream.

463

E. COLI INFECTION Escherichia coli (E. coli) is a bacterium that can reproduce at an exponential rate. The E. coli reproduce by dividing. A small number of E. coli bacteria in the large intestine of a human can trigger a serious infection within a few hours. Consider a particular E. coli infection that starts with 100 E. coli bacteria. Each bacterium splits into two parts every half hour. Assuming none of the bacteria die, the size of the E. coli population after t hours is given by P共t兲苷 100 22t, where 0 t 16. a. Find P共3兲 and P共6兲. b. Use a graphing utility to find the time, to the nearest tenth of an hour, it takes for the E. coli population to number 1 billion.

a. Find the amount of medication, to the nearest milligram, in the patient’s bloodstream after 45 minutes.

RADIATION Lead shielding is used to contain radiation. The percentage of a certain radiation that can penetrate x millimeters of lead shielding is given by I共x兲苷 100e1.5x.

b. Use a graphing utility to determine how long it will take, to the nearest minute, for the amount of medication in the patient’s bloodstream to reach 50 milligrams.

a. What percentage of radiation, to the nearest tenth of a percent, will penetrate a lead shield that is 1 millimeter thick?

49. DEMAND FOR A PRODUCT The demand d for a specific product, in items per month, is given by

b. How many millimeters of lead shielding are required so that less than 0.05% of the radiation penetrates the shielding? Round to the nearest millimeter.

52.

d共p兲苷 25 880e0.18p where p is the price, in dollars, of the product. a. What will be the monthly demand, to the nearest unit, when the price of the product is $8 and when the price is $18? b. What will happen to the demand as the price increases without bound? 50. SALES The monthly income I, in dollars, from a new product is given by I共t兲苷 24,000 22,000e0.005t where t is the time, in months, since the product was first put on the market.

53.

PHOTOCHROMATIC EYEGLASS LENSES Photochromatic eyeglass lenses contain molecules of silver chloride or silver halide. These molecules are transparent in the absence of UV rays. UV rays are normally absent in artificial lighting. However, when the lenses are exposed to UV rays, as in direct sunlight, the molecules take on a new molecular structure, which causes the lenses to darken. The number of molecules affected varies with the intensity of the UV rays. The intensity of UV rays is measured using a scale called the UV index. On this scale, a value near 0 indicates a low UV intensity and a value near 10 indicates a high UV intensity.

464

Chapter 7

Exponential and Logarithmic Functions

For the photochromatic lenses shown below, the function P(x) 苷 (0.9)x models the transparency P of the lenses as a function of the UV index x.

C PHOTOCHROMATIC PHOTOCHROMATIC

C PHOTOCHROMATIC PHOTOCHROMATIC

C PHOTOCHROMATIC PHOTOCHROMATIC

55.

UV index, 0 lens transparency, 100%

UV index, 5 lens transparency, 59.0%

UV index, 9 lens transparency, 38.7%

a. Find the transparency of these lenses, to the nearest tenth of a percent, when they are exposed to light rays with a UV index of 3.5.

FISH POPULATION The number of bass in a lake is given by P共t兲苷

PAY IT FORMODEL In the movie Pay It Forward, Trevor McKinney, played by Haley Joel Osment, is given a school assignment to “think of an idea to change the world—and then put it into action.” In response to this assignment, Trevor develops a pay it forward project. In this project, anyone who benefits from another person’s good deed must do a good deed for three additional people. Each of these three people is then obligated to do a good deed for another three people, and so on. The following diagram shows the number of people who have been a beneficiary of a good deed after 1 round and after 2 rounds of this project.

Three beneficiaries after one round.

b. What is the UV index of light rays that cause these photochromatic lenses to have a transparency of 45%? Round to the nearest tenth. 54.

THE

WARD

A total of 12 beneficiaries after two rounds (3 + 9 = 12).

3600 1 7e0.05t

where t is the number of months that have passed since the lake was stocked with bass.

A mathematical model for the number of pay-it-forward 3n1 3 beneficiaries after n rounds is given by B共n兲苷 . 2 Use this model to determine a. the number of beneficiaries after 5 rounds and after 10 rounds. Assume that no person is a beneficiary of more than one good deed. b. how many rounds are required to produce at least 2 million beneficiaries. 56.

a. How many bass were in the lake immediately after it was stocked? b. How many bass were in the lake 1 year after the lake was stocked? Round to the nearest bass. c. What will happen to the bass population as t increases without bound?

INTENSITY OF LIGHT The percent I共x兲 of the original intensity of light striking the surface of a lake that is available x feet below the surface of the lake is given by I共x兲苷 100e0.95x. a. What percentage of the light, to the nearest tenth of a percent, is available 2 feet below the surface of the lake? b. At what depth, to the nearest hundredth of a foot, is the intensity of the light one-half the intensity at the surface?

7.1 57.

Exponential Functions and Their Applications

A TEMPERATURE MODEL A cup of coffee is heated to 180 F and placed in a room that maintains a temperature of 65 F. The temperature of the coffee after t minutes is given by T共t兲苷 65 115e0.042t.

59.

465

MUSICAL SCALES Starting on the left side of a standard 88-key piano, the frequency, in vibrations per second, of the nth note is given by f 共n兲苷共27.5兲2共n1兲/12. Symphony #9

by L. von Beethoven

a. Find the temperature, to the nearest degree, of the coffee 10 minutes after it is placed in the room. b. Use a graphing utility to determine when, to the nearest tenth of a minute, the temperature of the coffee will reach 100 F. 58.

Middle C

A TEMPERATURE MODEL Soup that is at a temperature of 170 F is poured into a bowl in a room that maintains a constant temperature. The temperature of the soup decreases according to the model given by T共t兲苷 75 95e0.12t, where t is time in minutes after the soup is poured.

D

E

a. Using this formula, determine the frequency, to the nearest hundredth of a vibration per second, of middle C, key number 40 on an 88-key piano. b. Is the difference in frequency between middle C (key number 40) and D (key number 42) the same as the difference in frequency between D (key number 42) and E (key number 44)? Explain.

a. What is the temperature, to the nearest tenth of a degree, of the soup after 2 minutes? b. A certain customer prefers soup at a temperature of 110 F. How many minutes, to the nearest 0.1 minute, after the soup is poured does the soup reach that temperature? c. What is the temperature of the room?

Connecting Concepts 66. f 共x兲苷兹1 e x

60. Verify that the hyperbolic cosine function ex ex is an even function. cosh共x兲苷 2 61. Verify that the hyperbolic sine function sinh共x兲苷

68. Let h(x) 苷 2(x 4). Express h as the composition of an exponential function f and a polynomial function g. 2

ex ex 2

is an odd function. 62. Graph g共x兲苷 10 x, and then sketch the graph of g reflected across the line given by y 苷 x. 63. Graph f 共x兲苷 e x, and then sketch the graph of f reflected across the line given by y 苷 x. In Exercises 64 to 67, determine the domain of the given function. Write the domain using interval notation.

64. f 共x兲苷

e x ex e x ex

65. f 共x兲苷

e 兩 x兩 1 ex

67. f 共x兲苷兹e x ex

69. Let h(x) 苷 e(2x5). Express h as the composition of an exponential function f and a polynomial function g. 70.

Explain why the graph of f(x) 苷

ex ex 2

can be produced by plotting the average height of g(x) 苷 ex and h(x) 苷 ex for each value of x.

466

Chapter 7

Exponential and Logarithmic Functions

Projects 1.

THE SAINT LOUIS GATEWAY ARCH The Gateway Arch in Saint Louis was designed in the shape of an inverted catenary, as shown by the red curve in the drawing below. The Gateway Arch is one of the largest optical illusions ever created. As you look at the arch (and its basic shape defined by the catenary curve), it appears to be much taller than it is wide. However, this is not the case. The height of the catenary is given by

冉

h共x兲苷 693.8597 68.7672

2.

冊

e 0.0100333x e0.0100333x 2

where x and h共x兲 are measured in feet and x 苷 0 represents the position at ground level that is directly below the highest point of the catenary.

AN EXPONENTIAL REWARD According to legend, when Sissa Ben Dahir of India invented the game of chess, King Shirham was so impressed with the game that he summoned the game’s inventor and offered him the reward of his choosing. The inventor pointed to the chessboard and requested, for his reward, one grain of wheat on the first square, two grains of wheat on the second square, four grains on the third square, eight grains on the fourth square, and so on for all 64 squares on the chessboard. The king considered this a very modest reward and said he would grant the inventor’s wish. The following table shows how many grains of wheat are on each of the first six squares and the total number of grains of wheat needed to cover squares 1 to n for n 6.

y

x

a. Use a graphing utility to graph h共x兲. b. Use your graph to find the height of the catenary for x 苷 0, 100, 200, and 299 feet. Round each result to the nearest tenth of a foot. c. What is the width of the catenary at ground level and what is the maximum height of the catenary? Round each result to the nearest tenth of a foot. d. By how much does the maximum height of the catenary exceed its width at ground level? Round to the nearest tenth of a foot.

Square number, n

Number of grains of wheat on square n

Total number of grains of wheat on squares 1 through n

1

1

1

2

2

3

3

4

7

4

8

15

5

16

31

6

32

63

a. If all 64 squares of the chessboard are piled with wheat as requested by Sissa Ben Dahir, how many grains of wheat are on the board? b. A grain of wheat weighs approximately 0.000008 kilogram. Find the total weight of the wheat requested by Sissa Ben Dahir. c. In a recent year, a total of 6.5 10 8 metric tons of wheat were produced in the world. At this level, how many years, to the nearest year, of wheat production would be required to fill the request of Sissa Ben Dahir? One metric ton equals 1000 kilograms.

7.2

Section 7.2 ■ ■

■

■

Logarithmic Functions Graphs of Logarithmic Functions Domains of Logarithmic Functions Common and Natural Logarithms

Logarithmic Functions and Their Applications

467

Logarithmic Functions and Their Applications P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A37.

PS1. If 2x 苷 16, determine the value of x. [7.1] PS2. If 3x 苷

1 , determine the value of x. [7.1] 27

PS3. If x 4 苷 625, determine the value of x. [7.1] PS4. Find the inverse of f 共x兲苷

2x . [1.6] x3

PS5. State the domain of g共x兲苷兹x 2 . [1.3] PS6. If the range of h共x兲 is the set of all positive real numbers, then what is the domain of h1共x兲? [1.6]

Logarithmic Functions Every exponential function of the form g共x兲苷 b x is a one-to-one function and therefore has an inverse function. Sometimes we can determine the inverse of a function represented by an equation by interchanging the variables of its equation and then solving for the dependent variable. If we attempt to use this procedure for g共x兲苷 b x, we obtain

Math Matters

g共x兲苷 b x y 苷 bx x 苷 by

• Interchange the variables.

None of our previous methods can be used to solve the equation x 苷 b y for the exponent y. Thus we need to develop a new procedure. One method would be to merely write y 苷 the power of b that produces x Although this would work, it is not very concise. We need a compact notation to represent “y is the power of b that produces x.” This more compact notation is given in the following definition.

Logarithms were developed by John Napier (1550–1617) as a means of simplifying the calculations of astronomers. One of his ideas was to devise a method by which the product of two numbers could be determined by performing an addition.

Definition of a Logarithm and a Logarithmic Function If x 0 and b is a positive constant 共b 苷 1兲, then y 苷 logb x if and only if by 苷 x The notation logb x is read “the logarithm (or log) base b of x.” The function defined by f共x兲苷 logb x is a logarithmic function with base b. This function is the inverse of the exponential function g共x兲苷 b x.

468

Chapter 7

Exponential and Logarithmic Functions It is essential to remember that f共x兲苷 logb x is the inverse function of g共x兲苷 b x. Because these functions are inverses and because functions that are inverses have the property that f共 g共x兲兲苷 x and g共 f共x兲兲苷 x, we have the following important relationships.

Composition of Logarithmic and Exponential Functions Let g共x兲苷 b x and f共x兲苷 logb x 共x 0, b 0, b 苷 1兲. Then g共 f共x兲兲苷 b logb x 苷 x

take note

As an example of these relationships, let g共x兲苷 2x and f共x兲苷 log2 x. Then 2log2 x 苷 x

The notation logb x replaces the phrase “the power of b that

abbreviated 3 苷 log2 8. In your

log2 2x 苷 x

and

The equations y 苷 logb x

produces x.” For instance, “3 is the power of 2 that produces 8” is

f共 g共x兲兲苷 logb b x 苷 x

and

by 苷 x

and

are different ways of expressing the same concept.

work with logarithms, remember that a logarithm is an exponent.

Definition of Exponential Form and Logarithmic Form The exponential form of y 苷 logb x is b y 苷 x. The logarithmic form of b y 苷 x is y 苷 logb x.

These concepts are illustrated in the next two examples.

EXAMPLE 1

»

Change from Logarithmic to Exponential Form

Write each equation in its exponential form. 3 苷 log 2 8

a.

b.

2 苷 log10共x 5兲

c.

log e x 苷 4

Solution Use the definition y 苷 logb x if and only if b y 苷 x. Logarithms are exponents.

3 苷 log 2 8

a.

if and only if

23 苷 8

Base

b.

2 苷 log10共x 5兲 if and only if 10 2 苷 x 5.

c.

log e x 苷 4 if and only if e 4 苷 x.

d.

log b b3 苷 3 if and only if b 3 苷 b 3.

»

Try Exercise 4, page 476

d.

log b b3 苷 3

7.2

Logarithmic Functions and Their Applications

EXAMPLE 2

»

469

Change from Exponential to Logarithmic Form

Write each equation in its logarithmic form. 32 苷 9

a.

b.

53 苷 x

ab 苷 c

c.

d.

blogb 5 苷 5

Solution The logarithmic form of b y 苷 x is y 苷 log b x. Exponent

a.

3 苷9

b.

5 苷 x if and only if 3 苷 log 5 x.

c.

ab 苷 c if and only if b 苷 log a c.

d.

blogb 5 苷 5 if and only if log b 5 苷 log b 5.

2

2 苷 log 3 9

if and only if Base

3

»

Try Exercise 14, page 476

The definition of a logarithm and the definition of an inverse function can be used to establish many properties of logarithms. For instance: ■

log b b 苷 1 because b 苷 b1.

■

log b 1 苷 0 because 1 苷 b0.

■

log b共b x 兲苷 x because b x 苷 b x.

■

blogb x 苷 x because f共x兲苷 log b x and g共x兲苷 b x are inverse functions. Thus g关 f共x兲兴苷 x.

We will refer to the preceding properties as the basic logarithmic properties. Basic Logarithmic Properties 1. log b b 苷 1

EXAMPLE 3

2.

»

log b 1 苷 0

3.

log b共b x 兲苷 x

log 8 1

b. log 5 5

c.

log 2共24 兲

Solution a.

By Property 2, log8 1 苷 0.

b.

By Property 1, log 5 5 苷 1.

c.

By Property 3, log 2共24 兲苷 4.

d.

By Property 4, 3log 3 7 苷 7.

»

blogb x 苷 x

Apply the Basic Logarithmic Properties

Evaluate each of the following logarithms. a.

4.

Try Exercise 32, page 476

d.

3log 3 7

470

Chapter 7

Exponential and Logarithmic Functions Some logarithms can be evaluated just by remembering that a logarithm is an exponent. For instance, log5 25 equals 2 because the base 5 raised to the second power equals 25. ■

log10 100 苷 2 because 10 2 苷 100.

■

log 4 64 苷 3 because 43 苷 64.

■

log 7

1 1 1 苷 2 because 7 2 苷 2 苷 . 49 7 49

What is the value of log5 625?

QUESTION

Graphs of Logarithmic Functions y

g(x) = 2

8 6

x

y=x

4

Table 7.7

2 −4

−2

Because f共x兲苷 log b x is the inverse function of g共x兲苷 b x, the graph of f is a reflection of the graph of g across the line given by y 苷 x. The graph of g共x兲苷 2x is shown in Figure 7.13. Table 7.7 below shows some of the ordered pairs of the graph of g.

x

3

2

1

0

1

2

3

g (x) 2 x

1 8

1 4

1 2

1

2

4

8

f(x) = log2 x 2

4

6

8 x

−2 −4

Figure 7.13

The graph of the inverse of g, which is f共x兲苷 log 2 x, is also shown in Figure 7.13. Some of the ordered pairs of f are shown in Table 7.8. Note that if 共x, y兲 is a point on the graph of g, then 共 y, x兲 is a point on the graph of f. Also notice that the graph of f is a reflection of the graph of g across the line given by y 苷 x. Table 7.8 x

1 8

1 4

1 2

1

2

4

8

f (x) log2 x

3

2

1

0

1

2

3

The graph of a logarithmic function can be drawn by first rewriting the function in its exponential form. This procedure is illustrated in Example 4.

ANSWER

log 5 625 苷 4 because 54 苷 625.

7.2

471

Logarithmic Functions and Their Applications

EXAMPLE 4

»

Graph a Logarithmic Function

Graph f共x兲苷 log3 x.

Solution To graph f共x兲苷 log3 x, consider the equivalent exponential equation x 苷 3 y. Because this equation is solved for x, choose values of y and calculate the corresponding values of x, as shown in Table 7.9. Table 7.9 x 3y

1 9

1 3

1

3

9

y

2

1

0

1

2

Now plot the ordered pairs and connect the points with a smooth curve, as shown in Figure 7.14. y 3

f(x) = log3 x

2 1 2

−1

4

6

8

10 x

−2 −3

Figure 7.14

»

Try Exercise 44, page 476

y

We can use a similar procedure to draw the graph of a logarithmic function with a fractional base. For instance, consider y 苷 log2/3 x. Rewriting this in expo2 y nential form gives us 苷 x. Choose values of y and calculate the correspon3 ding x values. See Table 7.10. Plot the points corresponding to the ordered pairs 共x, y兲, and then draw a smooth curve through the points, as shown in Figure 7.15.

4

冉冊

2

2

4

x

−2

Table 7.10 −4

x y = log 2/3 x

Figure 7.15

冉冊冉冊 2 3

y

y

2 3

2

苷

2

9 4

冉冊 2 3

1

苷

1

3 2

冉冊 2 3

0

0

苷1

冉冊 2 3

1

1

苷

2 3

冉冊 2 3

2

2

苷

4 9

472

Chapter 7

Exponential and Logarithmic Functions Properties of f (x) log b x For all positive real numbers b, b 苷 1, the function f共x兲苷 log b x has the following properties: 1. The domain of f consists of the set of positive real numbers, and its range consists of the set of all real numbers. 2. The graph of f has an x-intercept of 共1, 0兲 and passes through 共b, 1兲. 3. If b 1, f is an increasing function and its graph is asymptotic to the negative y-axis. [As x l , f共x兲 l , and as x l 0 from the right, f共x兲 l .] See Figure 7.16a. 4. If 0 b 1, f is a decreasing function and its graph is asymptotic to the positive y-axis. [As x l , f共x兲 l , and as x l 0 from the right, f共x兲 l .] See Figure 7.16b. y 3

y 3

3

(b , 3) (b 2, 2)

(b 1, 1) b −1

(b, 1)

(1, 0) 1

b

b2

b3

x

(1, 0)

b −2 (b −1, −1)

(b −1, −1)

(b −2, −2)

a. f (x) = logb x, b > 1

b −3

x

(b −3, −3)

b. f (x) = logb x, 0 < b < 1

Figure 7.16

Domains of Logarithmic Functions The function f共x兲苷 logb x has as its domain the set of positive real numbers. The function f共x兲苷 logb共 g共x兲兲 has as its domain the set of all x for which g共x兲 0. To determine the domain of a function such as f共x兲苷 logb共 g共x兲兲, we must determine the values of x that make g共x兲 positive. This process is illustrated in Example 5.

EXAMPLE 5

»

Find the Domain of a Logarithmic Function

Find the domain of each of the following logarithmic functions. a.

f共x兲苷 log6共x 3兲

b.

F共x兲苷 log 2兩x 2兩

c.

冉冊

R共x兲苷 log 5

x 8x

Solution a.

Solving 共x 3兲 0 for x gives us x 3. The domain of f consists of all real numbers greater than 3. In interval notation, the domain is 共3, 兲.

b.

The solution set of 兩x 2兩 0 consists of all real numbers x except x 苷 2. The domain of F consists of all real numbers x 苷 2. In interval notation, the domain is 共, 2兲共2, 兲.

7.2

473

Logarithmic Functions and Their Applications

冉冊

x 0 yields the set of all real numbers x between 0 and 8. 8x The domain of R is all real numbers x such that 0 x 8. In interval notation, the domain is 共0, 8兲.

Solving

c.

»

Try Exercise 52, page 476

Some logarithmic functions can be graphed by using horizontal and/or vertical translations of a previously drawn graph. EXAMPLE 6

Graph.

»

Use Translations to Graph Logarithmic Functions

f共x兲苷 log4共x 3兲

a.

b. f共x兲苷 log4 x 3

Solution a.

The graph of f共x兲苷 log4共x 3兲 can be obtained by shifting the graph of g共x兲苷 log4 x to the left 3 units. See Figure 7.17. Note that the domain of f consists of all real numbers x greater than 3 because x 3 0 for x 3. The graph of f is asymptotic to the vertical line x 苷 3.

b.

The graph of f共x兲苷 log4 x 3 can be obtained by shifting the graph of g共x兲苷 log4 x upward 3 units. See Figure 7.18. y

f(x) = log4 (x + 3) y

4 −3

−1

1

−2

3

x

f(x) = log4 x + 3

2

g(x) = log 4 x

2 −2

4

6

8

x

g(x) = log4 x

Figure 7.17

Figure 7.18

»

Try Exercise 66, page 476

Common and Natural Logarithms Two of the most frequently used logarithmic functions are common logarithms, which have base 10, and natural logarithms, which have base e (the base of the natural exponential function). Definition of Common and Natural Logarithms The function defined by f共x兲苷 log10 x is called the common logarithmic function. It is customarily written without stating the base as f共x兲苷 log x. The function defined by f共x兲苷 log e x is called the natural logarithmic function. It is customarily written as f共x兲苷 ln x.

474

Chapter 7

Exponential and Logarithmic Functions Most scientific or graphing calculators have a LOG key for evaluating common logarithms and an LN key to evaluate natural logarithms. For instance, using a graphing calculator, log 24 ⬇ 1.3802112

ln 81 ⬇ 4.3944492

and

The graphs of f共x兲苷 log x and f共x兲苷 ln x can be drawn using the same techniques we used to draw the graphs in the preceding examples. However, these graphs also can be produced with a graphing calculator by entering log x and ln x into the Y苷 menu. See Figure 7.19 and Figure 7.20.

Plot1 Plot2 Plot3 \Y 1 = log(X) \Y2 = ln(X) \ WINDOW \ Xmin=0 \ Xmax=9.4 \ Xscl=1 \ Ymin=-3 Ymax=3 Yscl=1 Xres=1

3 f(x) = ln x

9.4

0 f(x) = log x

−3

Figure 7.19

Figure 7.20

Observe that each graph passes through 共1, 0兲. Also note that as x l 0 from the right, the functional values f共x兲 l . Thus the y-axis is a vertical asymptote for each of the graphs. The domain of both f共x兲苷 log x and f共x兲苷 ln x is the set of positive real numbers. Each of these functions has a range consisting of the set of real numbers. Many applications can be modeled by logarithmic functions.

EXAMPLE 7

»

Applied Physiology

In the study The Pace of Life, M. H. Bornstein and H. G. Bornstein (Nature, Vol. 259, pp. 557–558, 1976) reported that as the population of a city increases, the average walking speed of a pedestrian also increases. An approximate relation between the average pedestrian walking speed s, in miles per hour, and the population x, in thousands, of a city is given by the function s(x) 苷 0.37 ln x 0.05 a.

Determine the average walking speed, to the nearest tenth of a mile per hour, in San Francisco, which has a population of 780,000, and in Round Rock, Texas, which has a population of 62,000.

b.

Estimate the population of a city for which the average pedestrian walking speed is 3.1 miles per hour. Round to the nearest hundred-thousand.

7.2

Logarithmic Functions and Their Applications

475

Solution

Math Matters Although logarithms were originally developed to assist with computations, logarithmic functions have a much broader use today. They are often used in such disciplines as geology, acoustics, chemistry, physics, and economics, to name a few.

The population of San Francisco, in thousands, is 780.

a.

s(x) 苷 0.37 ln x 0.05 s(780) 苷 0.37 ln 780 0.05 ⬇ 2.5

• Substitute 780 for x. • Use a calculator to evaluate.

The average walking speed in San Francisco is about 2.5 miles per hour. The population of Round Rock, in thousands, is 62. s(x) 苷 0.37 ln x 0.05 s(62) 苷 0.37 ln 62 0.05 ⬇ 1.6

• Substitute 62 for x. • Use a calculator to evaluate.

The average walking speed in Round Rock is about 1.6 miles per hour. Graph s(x) 苷 0.37 ln x 0.05 and s 苷 3.1 in the same viewing window.

b.

5

0

Intersection X=3801.8507 Y=3.1

−1.5

10,000

Xscl = 1000 Yscl = 1

The x value of the intersection point represents the population in thousands. The function indicates that a city with an average pedestrian walking speed of 3.1 miles per hour should have a population of about 3,800,000.

»

Try Exercise 86, page 477

Topics for Discussion 1. If m n, must log b m log b n? 2. For what values of x is ln x log x? 3. What is the domain of f共x兲苷 log共x 2 1兲? Explain why the graph of f does not have a vertical asymptote. 4.

The subtraction 3 5 does not have an answer if we require that the answer be positive. Keep this idea in mind as you work the rest of this discussion exercise. Press the MODE key of a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, and choose “Real” from the menu. Now use the calculator to evaluate log共2兲. What output is given by the calculator? Press the MODE key, and choose “a bi” from the menu. Now use the calculator to evaluate log共2兲. What output is given by the calculator? Explain why the output is different for these two evaluations.

476

Chapter 7

Exponential and Logarithmic Functions

Exercise Set 7.2 In Exercises 1 to 12, write each equation in its exponential form.

In Exercises 43 to 50, graph each function by using its exponential form.

1. 1 苷 log 10

2. 4 苷 log 10,000

43. f 共x兲苷 log 4 x

44. f 共x兲苷 log6 x

3. 2 苷 log8 64

4. 3 苷 log4 64

45. f 共x兲苷 log12 x

46. f 共x兲苷 log 8 x

5. 0 苷 log7 x

1 6. 4 苷 log3 81

47. f 共x兲苷 log1/2 x

48. f 共x兲苷 log1/4 x

7. ln x 苷 4

8. ln e2 苷 2

49. f 共x兲苷 log 5/2 x

50. f 共x兲苷 log 7/3 x

9. ln 1 苷 0

10. ln x 苷 3

11. 2 苷 log(3x 1)

12.

冉冊

1 x1 苷 ln 3 x2

In Exercises 13 to 24, write each equation in its logarithmic form. Assume y > 0 and b > 0.

13. 3 苷 9

14. 5 苷 125

2

3

1 15. 42 苷 16

16. 10 0 苷 1

17. bx 苷 y

18. 2x 苷 y

19. y 苷 ex

20. 51 苷 5

21. 100 苷 10 2

22. 24 苷

23. e2 苷 x 5

24. 3x 苷 47

1 16

In Exercises 25 to 42, evaluate each logarithm. Do not use a calculator.

25. log4 16 27. log3

1 243

29. ln e3 31. log

1 100

In Exercises 51 to 64, find the domain of the function. Write the domain using interval notation.

51. f 共x兲苷 log 5共x 3兲

52. k共x兲苷 log4共5 x兲

53. k共x兲苷 log 2/3共11 x兲

54. H共x兲苷 log1/4共x 2 1兲

55. P共x兲苷 ln共x 2 4兲

56. J共x兲苷 ln

冉冊

57. h共x兲苷 ln

60. s共x兲苷 log 7共x 2 7x 10兲

61. g(x) 苷 log 兹2x 11

62. m(x) 苷 log 兩4x 8兩

63. t(x) 苷 2 ln(3x 7)

64. v(x) 苷 ln(x 4)2

In Exercises 65 to 72, use translations of the graphs in Exercises 43 to 50 to produce the graph of the given function.

65. f 共x兲苷 log 4共x 3兲

66. f 共x兲苷 log6共x 3兲

67. f 共x兲苷 log12 x 2

68. f 共x兲苷 log 8 x 4

69. f 共x兲苷 3 log1/2 x

70. f 共x兲苷 2 log1/4 x

28. logb 1

71. f 共x兲苷 1 log 5/2共x 4兲

72. f 共x兲苷 log 7/3共x 3兲 1

30. logb b

73. Examine the following four functions and the graphs labeled a, b, c, and d. For each graph, determine which function has been graphed.

100 9

34. log0.3

35. 4 log 1000

36. log5 1252

37. 2 log7 2401

38. 3 log11 161,051 3

40. log6兹 36 3

41. 5 log13 兹 169

58. R共x兲苷 ln共x 4 x 2 兲

8 26. log3/2 27

33. log0.5 16

5

x2 x4

x3 x

59. N共x兲苷 log 2共x 3 x兲

32. log 1,000,000

39. log3兹 9

冉冊

7

42. 2 log7 兹 343

a.

f共x兲苷 log5共x 2兲

g共x兲苷 2 log5 x

h共x兲苷 log5共x兲

k共x兲苷 log5共x 3兲 b.

y 4

y 4

4 x

−4 −4

4 −4

x

7.2 c.

d.

y

86. AVERAGE TYPING SPEED The following function models the average typing speed S, in words per minute, of a student who has been typing for t months.

y

4

4

4 x

−4

S共t兲苷 5 29 ln共t 1兲,

4 x

−4

−4

f共x兲苷 ln x 3

g共x兲苷 ln共x 3兲

h共x兲苷 ln共3 x兲

k共x兲苷 ln共x兲

b.

y

b.

87.

y

4

0 t 16

a. What was the student’s average typing speed, to the nearest word per minute, when the student first started to type? What was the student’s average typing speed, to the nearest word per minute, after 3 months?

−4

74. Examine the following four functions and the graphs labeled a, b, c, and d. For each graph, determine which function has been graphed.

a.

4

Use a graph of S to determine how long, to the nearest tenth of a month, it will take the student to achieve an average typing speed of 65 words per minute. ADVERTISING COSTS AND SALES The function

冉

N共x兲苷 2750 180 ln 4 x

−4

c.

4 x

−4

−4

d.

y

a. Find the number of units that will be sold with advertising expenditures of $20,000, $40,000, and $60,000.

y 4

x

4 −4

In Exercises 75 to 84, use a graphing utility to graph the function.

75. f 共x兲苷 2 ln x

76. f 共x兲苷 log x

77. f 共x兲苷兩ln x兩

78. f 共x兲苷 ln 兩x兩

79. f 共x兲苷 log 兹 x

80. f 共x兲苷 ln 兹x

81. f 共x兲苷 log共x 10兲

82. f 共x兲苷 ln共x 3兲

83. f 共x兲苷 3 log 兩2x 10兩

84. f 共x兲苷

85.

b. How many units will be sold if the company does not pay to advertise the product?

4 x

−4 −4

3

冊

x 1 1000

models the relationship between the dollar amount x spent on advertising a product and the number of units N that a company can sell.

−4

4

477

Logarithmic Functions and Their Applications

1 ln 兩x 4兩 2

MONEY MARKET RATES The function r共t兲苷 0.69607 0.60781 ln t

MEDICINE In anesthesiology it is necessary to accurately estimate the body surface area of a patient. One formula for estimating body surface area (BSA) was developed by Edith Boyd (University of Minnesota Press, 1935). Her formula for the BSA (in square meters) of a patient of height H (in centimeters) and weight W (in grams) is BSA 0.0003207 H 0.3 W (0.72850.0188 log W) In Exercises 88 and 89, use Boyd’s formula to estimate the body surface area of a patient with the given weight and height. Round to the nearest hundredth of a square meter.

88. W 苷 110 pounds 共49,895.2 grams兲; H 苷 5 feet 4 inches 共162.56 centimeters兲

gives the annual interest rate r, as a percent, a bank will pay on its money market accounts, where t is the term (the time the money is invested) in months.

89. W 苷 180 pounds 共81,646.6 grams兲; H 苷 6 feet 1 inch 共185.42 centimeters兲

a. What interest rate, to the nearest tenth of a percent, will the bank pay on a money market account with a term of 9 months?

90.

b. What is the minimum number of complete months during which a person must invest to receive an interest rate of at least 3%?

ASTRONOMY Astronomers measure the apparent brightness of a star by a unit called the apparent magnitude. This unit was created in the second century B.C. when the Greek astronomer Hipparchus classified the relative brightness of several stars. In his list he assigned the number 1 to the stars that appeared to be the brightest (Sirius, Vega, and Deneb). They are

478

Chapter 7

Exponential and Logarithmic Functions

first-magnitude stars. Hipparchus assigned the number 2 to all the stars in the Big Dipper. They are secondmagnitude stars. The following table shows the relationship between a star’s brightness relative to a firstmagnitude star and the star’s apparent magnitude. Notice from the table that a first-magnitude star appears to be about 2.51 times as bright as a second-magnitude star.

c. Which star appears brighter: a star with an apparent magnitude of 12 or a star with an apparent magnitude of 15? d. Is M共x兲 an increasing function or a decreasing function? 91.

NUMBER OF DIGITS IN bx An engineer has determined that the number of digits N in the expansion of bx, where both b and x are positive integers, is N 苷 int共x log b兲 1, where int共x log b兲 denotes the greatest integer of x log b. (Note: See pages 41–44 for information on the greatest integer function.)

Brightness, x relative to a first-magnitude star

Apparent magnitude M(x)

1

1

1 2.51

2

b. Find the number of digits in 3200.

1 1 ⬇ 6.31 2.512

3

c. Find the number of digits in 74005.

1 1 ⬇ 15.85 2.513

4

1 1 ⬇ 39.82 2.514

5

1 1 ⬇ 100 2.515

6

a. Because 210 苷 1024, we know that 210 has four digits. Use the equation N 苷 int共x log b兲 1 to verify this result.

d.

92.

The following logarithmic function gives the apparent magnitude M共x兲 of a star as a function of its brightness x. M共x兲苷 2.51 log x 1,

The largest known prime number as of November 17, 2003, was 220996011 1. Find the number of digits in this prime number. (Hint: Because 220996011 is not a power of 10, both 220996011 and 220996011 1 have the same number of digits.) 9

NUMBER OF DIGITS IN 9(9 ) A science teacher has offered 10 points extra credit to any student who will 9 write out all the digits in the expansion of 9(9 ). a. Use the formula from Exercise 91 to determine the number of digits in this number.

0x1

b. Assume that you can write 1000 digits per page and that 500 pages of paper are in a ream of paper. How many reams of paper, to the nearest tenth of a ream, are 9 required to write out the expansion of 9(9 )? Assume that you write on only one side of each page.

a. Use M共x兲 to find the apparent magnitude of a star that 1 is as bright as a first-magnitude star. Round to the 10 nearest hundredth. b. Find the approximate apparent magnitude of a star 1 that is as bright as a first-magnitude star. Round to 400 the nearest hundredth.

Connecting Concepts 93.

e x ex and 2 2 g共x兲苷 ln共 x 兹x 1 兲 on the same screen. Use a square viewing window. What appears to be the relationship between f and g? Use a graphing utility to graph f 共x兲苷

94.

e x ex for 2 2 x 0 and g共x兲苷 ln共 x 兹x 1 兲 for x 1 on the same screen. Use a square viewing window. What appears to be the relationship between f and g? Use a graphing utility to graph f 共x兲苷

7.2

479

Logarithmic Functions and Their Applications

e x ex 1x 1 and g共x兲苷 ln are e x ex 2 1x inverse functions. The domain of f is the set of all real numbers. The domain of g is 兵x 兩 1 x 1其. Use this information to determine the range of f and the range of g.

95. The functions f 共x兲苷

Use a graph of f 共x兲苷

96.

2 to determine the ex ex

domain and range of f.

Projects P(d)

BENFORD’S LAW The authors of this text know some interesting details about your finances. For instance, of the last 150 checks you have written, about 30% are for amounts that start with the number 1. Also, you have written about 3 times as many checks for amounts that start with the number 2 as you have for amounts that start with the number 7. We are sure of these results because of a mathematical formula known as Benford’s Law. This law was first discovered by the mathematician Simon Newcomb in 1881 and then rediscovered by the physicist Frank Benford in 1938. Benford’s Law states that the probability P that the first digit of a number selected from a wide range of numbers is d is given by

冉冊

P共d兲苷 log 1

1 d

a. Use Benford’s Law to complete the table below and the bar graph at the top of the next column.

d

冉冊

P共d 兲苷 log 1

1

0.301

2

0.176

3

0.125

4

1 d

0.30 The probability of d as the leading digit

1.

0.25 0.20 0.15 0.10 0.05 0.00

1

2

3

4

5

6

7

8

9

d

The leading digit d

Benford’s Law applies to many sets of data with a wide range. For instance, it applies to the populations of the cities in the U.S., the numbers of dollars in the savings accounts at your local bank, and the number of miles driven during a month by each person in a state. b. Use the table in a. to find the probability that in a U.S. city selected at random, the number of telephones in that city will be a number starting with 6. c. Use the table in a. to estimate how many times as many purchases you have made for dollar amounts that start with a 1 than for dollar amounts that start with a 9. d.

Explain why Benford’s Law would not apply to the set of all the ages, in years, of students at a local high school.

5 6 7 8 9

AN APPLICATION OF BENFORD’S LAW Benford’s Law has been used to identify fraudulent accountants. In most cases these accountants are unaware of Benford’s Law and have replaced valid numbers with numbers selected at random. Their numbers do not conform to Benford’s Law. Hence an audit is warranted.

480

Chapter 7

Exponential and Logarithmic Functions

Section 7.3 ■ ■ ■

Properties of Logarithms Change-of-Base Formula Logarithmic Scales

Properties of Logarithms and Logarithmic Scales P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A38. In Exercises PS1 to PS6, use a calculator to compare the values of the given expressions.

PS1. log 3 log 2; log 6 [7.2]

冉冊

PS2. ln 8 ln 3; ln

8 3

[7.2]

PS3. 3 log 4; log共43 兲 [7.2] PS4. 2 ln 5; ln共52 兲 [7.2] PS5. ln 5;

log 5 [7.2] log e

PS6. log 8;

ln 8 [7.2] ln 10

Properties of Logarithms In Section 7.2 we introduced the following basic properties of logarithms. log b b 苷 1 and log b 1 苷 0 Also, because exponential functions and logarithmic functions are inverses of each other, we observed the relationships log b共b x 兲苷 x and blogb x 苷 x

take note

We can use the properties of exponents to establish the following additional logarithmic properties.

Pay close attention to these properties. Note that logb共MN兲苷 logb M logb N and logb

M logb M 苷 N logb N

Properties of Logarithms In the following properties, b, M, and N are positive real numbers 共b 苷 1兲. Product property log b共MN兲苷 log b M log b N M 苷 log b M log b N N

Quotient property

log b

Power property

log b共M p 兲苷 p log b M

In fact, the expression logb 共M N兲

Logarithm-of-each-side property

M 苷 N implies log b M 苷 log b N

cannot be expanded at all.

One-to-one property

log b M 苷 log b N implies M 苷 N

Also, logb共M N兲苷 logb M logb N

7.3

Properties of Logarithms and Logarithmic Scales

481

Here is a proof of the product property. Proof: Let r 苷 logb M and s 苷 logb N. These equations can be written in exponential form as

M 苷 br

and

N 苷 bs

Now consider the product MN. MN 苷 brbs MN 苷 brs logb MN 苷 r s

• Product property of exponents

logb MN 苷 logb M logb N

• Substitute for r and s.

• Substitute for M and N.

• Write in logarithmic form.

◆

The last equation is our desired result.

The quotient property and the power property can be proved in a similar manner. See Exercises 87 and 88 on page 492. The properties of logarithms are often used to rewrite logarithmic expressions in an equivalent form. The process of using the product or quotient rules to rewrite a single logarithm as the sum or difference of two or more logarithms, or using the power property to rewrite logb共M p 兲 in its equivalent form p logb M, is called expanding the logarithmic expression. We illustrate this process in Example 1.

EXAMPLE 1

»

Expand Logarithmic Expressions

Use the properties of logarithms to expand the following logarithmic expressions. Assume all variable expressions represent positive real numbers. When possible, evaluate logarithmic expressions. log 5共xy 2 兲

a.

b.

冉冊

ln

e兹y z3

Solution log 5共xy 2 兲苷 log 5 x log 5 y 2 苷 log 5 x 2 log 5 y

a.

冉冊

ln

b.

»

e兹y z3

• Product property • Power property

苷 ln共e兹y兲 ln z 3

• Quotient property

苷 ln e ln 兹y ln z 3 苷 ln e ln y 1兾2 ln z 3 1 苷 ln e ln y 3 ln z 2 1 苷 1 ln y 3 ln z 2

• Product property

Try Exercise 2, page 489

• Write 兹y as y1/2. • Power property • Evaluate ln e.

482

Chapter 7

Exponential and Logarithmic Functions The properties of logarithms are also used to condense expressions that involve the sum or difference of logarithms into a single logarithm. For instance, we can use the product property to rewrite logb M logb N as logb共MN兲, and the quotient M . Before applying the product or property to rewrite logb M logb N as logb N quotient properties, use the power property to write all expressions of the form p logb M in their equivalent logb M p form. See Example 2. QUESTION

Does log 2 log 5 苷 1?

EXAMPLE 2

»

Condense Logarithmic Expressions

Use the properties of logarithms to rewrite each expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers. 2 ln x

a.

1 ln 共x 4兲 2

b. log5共x 2 4兲 3 log5 y log5共x 2兲2

Solution 2 ln x

a.

1 ln 共x 4兲苷 ln x 2 ln 共x 4兲1兾2 2

苷 ln 关x 共x 4兲 2

1兾2

兴

苷 ln 关x 兹共x 4兲兴 2

• Power property • Product property • Rewriting 共x 4兲1/2 as 兹x 4 is an optional step.

log5共x 2 4兲 3 log5 y log5共x 2兲2

b.

苷 log5 共x 2 4兲 log5 y 3 log5 共x 2兲2 苷关log5 共x 2 4兲 log5 y 3 兴 log5 共x 2兲2

• Power property

苷 log5 关共x 2 4兲 y 3 兴 log5 共x 2兲2

• Product property

苷 log5 苷 log5 苷 log5

冋冋冋

共x 4兲 y 共x 2兲2 2

册

• Order of Operations Agreement

3

• Quotient property

册

共x 2兲共x 2兲 y 3 共x 2兲2

• Factor.

共x 2兲 y 3 x2

• Simplify.

册

»

Try Exercise 18, page 490

Change-of-Base Formula Recall that to determine the value of y in log3 81 苷 y, we ask the question, “What power of 3 is equal to 81?” Because 34 苷 81, we have log3 81 苷 4. Now suppose ANSWER

Yes. By the product property, log 2 log 5 苷 log 共2 5兲苷 log 10 苷 1.

7.3

Properties of Logarithms and Logarithmic Scales

483

that we need to determine the value of log3 50. In this case we need to find the power of 3 that produces 50. Because 33 苷 27 and 34 苷 81, the value we are seeking is somewhere between 3 and 4. The following procedure can be used to produce an estimate of log3 50. The exponential form of log3 50 苷 y is 3 y 苷 50. Applying logarithmic properties gives us 3 y 苷 50 ln 3 y 苷 ln 50 • Logarithm-of-each-side property • Power property y ln 3 苷 ln 50 ln 50 • Solve for y. y苷 ⬇ 3.56088 ln 3 Thus log3 50 ⬇ 3.56088. In the above procedure we could just as well have used logarithms of any base and arrived at the same value. Thus any logarithm can be expressed in terms of logarithms of any base we wish. This general result is summarized in the following formula.

Change-of-Base Formula If x, a, and b are positive real numbers with a 苷 1 and b 苷 1, then log a x log b x 苷 log a b Because most calculators use only common logarithms (a 苷 10) or natural logarithms (a 苷 e), the change-of-base formula is used most often in the following form. If x and b are positive real numbers and b 苷 1, then log x ln x log b x 苷苷 log b ln b

EXAMPLE 3

»

Use the Change-of-Base Formula

Evaluate each logarithm. Round to the nearest ten thousandth. log 3 18

a.

b.

log12 400

take note If common logarithms had been used for the calculations in Example 3, the final results would have been the same. log 3 18 苷

log 18 ⬇ 2.6309 log 3

log 400 ⬇ 2.4111 log12 400 苷 log 12

Solution To approximate these logarithms, we may use the change-of-base formula with a 苷 10 or a 苷 e. For this example we choose to use the change-of-base formula with a 苷 e. That is, we will evaluate these logarithms by using the LN key on a scientific or graphing calculator. log3 18 苷

a.

»

ln 18 ⬇ 2.6309 ln 3

Try Exercise 34, page 490

b.

log12 400 苷

ln 400 ⬇ 2.4111 ln 12

484

Chapter 7

Exponential and Logarithmic Functions

y 4

f(x) = log3(2x + 3)

2

−2

2

4

x

The change-of-base formula and a graphing calculator can be used to graph logarithmic functions that have a base other than 10 or e. For instance, to graph f共x兲苷 log3共2x 3兲, we rewrite the function in terms of base 10 or base e. Using log共2x 3兲 base 10 logarithms, we have f共x兲苷 log3共2x 3兲苷 . The graph is log 3 shown in Figure 7.21.

−2

EXAMPLE 4

−4

Figure 7.21

»

Use the Change-of-Base Formula to Graph a Logarithmic Function

Graph f共x兲苷 log2 兩x 3兩.

Solution Rewrite f using the change-of-base formula. We will use the natural logarithm function; however, the common logarithm function could be used instead. f共x兲苷 log2 兩x 3兩苷 4

ln 兩x 3兩 into Y1. Note that ln 2 the domain of f (x) log 2 兩x 3兩 is

• Enter

f(x) = log2|x − 3| − 2.7

ln 兩x 3兩 ln 2

6.7

all real numbers except 3 because 兩x 3兩 0 when x 3 and 兩x 3兩 is positive for all other values of x.

−4

»

Try Exercise 46, page 490

Logarithmic Scales Logarithmic functions are often used to scale very large (or very small) numbers into numbers that are easier to comprehend. For instance, the Richter scale magnitude of an earthquake uses a logarithmic function to convert the intensity of the earthquake’s shock waves I into a number M, which for most earthquakes is in the range of 0 to 10. The intensity I of an earthquake is often given in terms of the constant I0, where I0 is the intensity of the smallest earthquake (called a zero-level earthquake) that can be measured on a seismograph near the earthquake’s epicenter. The following formula is used to compute the Richter scale magnitude of an earthquake.

Math Matters The Richter scale was created by the seismologist Charles F. Richter in 1935. Notice that a tenfold increase in the intensity level of an earthquake increases the Richter scale magnitude of the earthquake by only 1.

The Richter Scale Magnitude of an Earthquake An earthquake with an intensity of I has a Richter scale magnitude of I M 苷 log I0 where I0 is the measure of the intensity of a zero-level earthquake.

冉冊

7.3

485

Properties of Logarithms and Logarithmic Scales

EXAMPLE 5

»

Determine the Magnitude of an Earthquake

Find the Richter scale magnitude (to the nearest 0.1) of the 1999 Joshua Tree, California, earthquake that had an intensity of I 苷 12,589,254I0.

take note

Solution

Notice in Example 5 that we

冉冊冉

M 苷 log

didn’t need to know the value of I0

冊

I 12,589,254I0 苷 log 苷 log共12,589,254兲 ⬇ 7.1 I0 I0

to determine the Richter scale

The 1999 Joshua Tree earthquake had a Richter scale magnitude of 7.1.

magnitude of the quake.

»

Try Exercise 78, page 491

If you know the Richter scale magnitude of an earthquake, you can determine the intensity of the earthquake.

EXAMPLE 6

»

Determine the Intensity of an Earthquake

Find the intensity of the 1999 Taiwan earthquake, which measured 7.6 on the Richter scale.

Solution

冉冊

log

I 苷 7.6 I0 I 苷 10 7.6 I0 I 苷 10 7.6I0 I ⬇ 39,810,717I0

• Write in exponential form. • Solve for I.

The 1999 Taiwan earthquake had an intensity that was approximately 39,811,000 times the intensity of a zero-level earthquake.

»

Try Exercise 80, page 491

In Example 7 we make use of the Richter scale magnitudes of two earthquakes to compare the intensities of the earthquakes.

EXAMPLE 7

»

Compare Earthquakes

The 1960 Chile earthquake had a Richter scale magnitude of 9.5. The 1989 San Francisco earthquake had a Richter scale magnitude of 7.1. Compare the intensities of the earthquakes. Continued 䉴

486

Chapter 7

Exponential and Logarithmic Functions

take note The results of Example 7 show that if an earthquake has a Richter

Solution Let I1 be the intensity of the Chilean earthquake and I2 the intensity of the San Francisco earthquake. Then

冉冊

scale magnitude of M 1 and a

log

smaller earthquake has a Richter scale magnitude of M 2, then the larger earthquake is 10 M1M2 times as intense as the smaller earthquake.

I1 苷 9.5 I0

and

冉冊

log

I2 苷 7.1 I0

I1 苷 10 9.5 I0

I2 苷 10 7.1 I0

I1 苷 10 9.5I0

I2 苷 10 7.1I0

To compare the intensities of the earthquakes, we compute the ratio I1兾I2. I1 10 9.5I0 10 9.5 苷苷苷 10 9.57.1 苷 10 2.4 ⬇ 251 I 2 10 7.1I0 10 7.1 The earthquake in Chile was approximately 251 times as intense as the San Francisco earthquake.

»

Try Exercise 82, page 491

Seismologists generally determine the Richter scale magnitude of an earthquake by examining a seismogram. See Figure 7.22. Arrival of first s-wave Arrival of first p-wave

Amplitude = 23 mm

24 seconds Time between s-wave and p-wave

Figure 7.22

The magnitude of an earthquake cannot be determined just by examining the amplitude of a seismogram because this amplitude decreases as the distance between the epicenter of the earthquake and the observation station increases. To account for the distance between the epicenter and the observation station, a seismologist examines a seismogram for both small waves called p-waves and larger waves called s-waves. The Richter scale magnitude M of the earthquake is a function of both the amplitude A of the s-waves and the difference in time t between the occurrence of the s-waves and the occurrence of the p-waves. In the 1950s, Charles Richter developed the following formula to determine the magnitude of an earthquake from the data in a seismogram.

7.3

Properties of Logarithms and Logarithmic Scales

487

Amplitude-Time-Difference Formula The Richter scale magnitude M of an earthquake is given by M 苷 log A 3 log 8t 2.92 where A is the amplitude, in millimeters, of the s-waves on a seismogram and t is the difference in time, in seconds, between the s-waves and the p-waves.

EXAMPLE 8

»

Determine the Magnitude of an Earthquake from Its Seismogram

Find the Richter scale magnitude of the earthquake that produced the seismogram in Figure 7.22.

Solution M 苷 log A 3 log 8t 2.92 苷 log 23 3 log关8 24兴 2.92

take note The Richter scale magnitude is

⬇ 1.36173 6.84990 2.92

usually rounded to the nearest

⬇ 5.3

tenth.

• Substitute 23 for A and 24 for t.

The earthquake had a magnitude of about 5.3 on the Richter scale.

»

Try Exercise 86, page 492

Logarithmic scales are also used in chemistry. One example concerns the pH of a liquid, which is a measure of the liquid’s acidity or alkalinity. (You may have tested the pH of the water in a swimming pool or an aquarium.) Pure water, which is considered neutral, has a pH of 7.0. The pH scale ranges from 0 to 14, with 0 corresponding to the most acidic solutions and 14 to the most alkaline. Lemon juice has a pH of about 2, whereas household ammonia measures about 11. Specifically, the pH of a solution is a function of the hydronium-ion concentration of the solution. Because the hydronium-ion concentration of a solution can be very small (with values such as 0.00000001), pH uses a logarithmic scale.

take note One mole is equivalent to 6.022 1023 ions.

Definition of the pH of a Solution The pH of a solution with a hydronium-ion concentration of H moles per liter is given by pH 苷 log关H 兴

488

Chapter 7

Exponential and Logarithmic Functions EXAMPLE 9

»

Find the pH of a Solution

Find the pH of each liquid. Round to the nearest tenth. a.

Orange juice with H 苷 2.8 10 4 mole per liter

b.

Milk with H 苷 3.97 10 7 mole per liter

c.

Rainwater with H 苷 6.31 10 5 mole per liter

d.

A baking soda solution with H 苷 3.98 10 9 mole per liter

Solution

Math Matters The pH scale was created by the Danish biochemist Søren Sørensen in 1909 to measure the acidity of water used in the brewing of beer. pH is an abbreviation for pondus hydrogenii, which translates as “potential hydrogen.”

a.

pH 苷 log关H 兴苷 log共2.8 10 4 兲 ⬇ 3.6 The orange juice has a pH of 3.6.

b.

pH 苷 log关H 兴苷 log共3.97 10 7 兲 ⬇ 6.4 The milk has a pH of 6.4.

c.

pH 苷 log关H 兴苷 log共6.31 10 5 兲 ⬇ 4.2 The rainwater has a pH of 4.2.

d.

pH 苷 log关H 兴苷 log共3.98 10 9 兲 ⬇ 8.4 The baking soda solution has a pH of 8.4.

»

Try Exercise 70, page 491

Figure 7.23 illustrates the pH scale, along with the corresponding hydroniumion concentrations. A solution on the left half of the scale, with a pH of less than 7, is an acid, and a solution on the right half of the scale is an alkaline solution or a base. Because the scale is logarithmic, a solution with a pH of 5 is 10 times more acidic than a solution with a pH of 6. From Example 9 we see that the orange juice, rainwater, and milk are acids, whereas the baking soda solution is a base.

Neutral Acidic pH

0

H+ 100

Basic

1

2

3

4

5

6

7

10−1

10−2

10−3

10−4

10−5

10−6

10−7

8

9

10

11

12

13

14

10−8

10−9

10−10

10−11

10−12

10−13

10−14

+

pH = −log[H ]

Figure 7.23

7.3

Properties of Logarithms and Logarithmic Scales

EXAMPLE 10

»

489

Find the Hydronium-Ion Concentration

A sample of blood has a pH of 7.3. Find the hydronium-ion concentration of the blood.

Solution pH 苷 log关H 兴 7.3 苷 log关H 兴 7.3 苷 log关H 兴 10 7.3 苷 H 5.0 10 8 ⬇ H

• Substitute 7.3 for pH. • Multiply both sides by 1. • Change to exponential form.

The hydronium-ion concentration of the blood is about 5.0 10 8 mole per liter.

»

Try Exercise 72, page 491

Topics for Discussion 1. The function f共x兲苷 log b x is defined only for x 0. Explain why this condition is imposed. 2. If p and q are positive numbers, explain why ln共 p q兲 isn’t normally equal to ln p ln q. 3. If f共x兲苷 log b x and f共c兲苷 f共d兲, can we conclude that c 苷 d? 4. Give examples of situations in which it is advantageous to use logarithmic scales.

Exercise Set 7.3 In Exercises 1 to 16, expand the given logarithmic expression. Assume all variable expressions represent positive real numbers. When possible, evaluate logarithmic expressions. Do not use a calculator.

1. log b共xyz兲

3. ln

x z4

5. log2

2. ln

兹xz y2

9. ln 共e2z兲

8. ln 兹 x 2 兹y 3

10. ln 共x 1兾2 y 2兾3 兲

冉冊

z3

3

兹xy

4. log 5 兹x y3

7. log 7

xy 2 z4

6. log b 共 x兹 y 兲 3

兹z 16y 3

11. log4

13. log 兹x兹z

14. ln

15. ln共兹 z兹e 兲

16. ln

3

冉冊冉冊冋册

12. log5

兹xz 4 125

3 2 x 兹

z2

x 2兹z y 3

490

Chapter 7

Exponential and Logarithmic Functions

In Exercises 17 to 32, write each expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.

41. log兹2 17

42. log兹3 5.5

43. logp e

44. logp 兹15

17. log共x 5兲 2 log x In Exercises 45 to 52, use a graphing utility and the change-of-base formula to graph the logarithmic function.

1 18. 3 log2 t log2 u 4 log2 v 3 19. ln共x 2 y 2 兲 ln共x y兲 20.

1 log8共x 5兲 3 log8 y 2

21. 3 log x

1 log y log共x 1兲 3

22. ln共xz兲 ln共 x兹y 兲 2 ln

y z

24. ln共y

z兲 ln z

47. g共x兲苷 log 8共x 3兲

48. t共x兲苷 log 9共5 x兲

49. h共x兲苷 log 3 共x 3兲2

50. J共x兲苷 log12共x兲

51. F共x兲苷 log5 兩x 2兩

52. n共x兲苷 log 2兹x 8

53. logb共x y兲苷 logb x logb y

1兾2

54. logb共xy兲苷 logb x logb y

25. 2共log6 x log6 y 兲 log6共x 2兲 2

26.

46. g共x兲苷 log8 共5 x兲

In Exercises 53 to 62, determine if the statement is true or false for all x w 0, y w 0. If it is false, write an example that disproves the statement.

23. log共xy 2 兲 log z 1兾2

45. f 共x兲苷 log 4 x

55. logb共xy兲苷 logb x logb y

1 log3 x log3 y 2 log3共x 2兲 2

56. logb x logb y 苷 logb x logb y 57. logb x logb y 苷 logb共x y兲,

27. 2 ln共x 4兲 ln x ln共x 2 3兲 28. log共3x兲共2 log x log y兲 29. ln共2x 5兲 ln y 2 ln z

58. logb 1 ln w 2

59.

30. logb x logb共y 3兲 logb共y 2兲 logb共y 2 5y 6兲 31. ln共x 2 9兲 2 ln共x 3兲 3 ln y

x logb x 苷 y logb y

logb x 苷 logb x logb y logb y

60. logb共x n 兲苷 n logb x 61. 共logb x兲n 苷 n logb x

32. logb共x 2 7x 12兲 2 logb共x 4兲 62. logb 兹x 苷 In Exercises 33 to 44, use the change-of-base formula to approximate the logarithm accurate to the nearest ten thousandth.

33. log 7 20

34. log 5 37

35. log11 8

36. log 50 22

37. log6

1 3

39. log9 兹17

38. log3

xy

7 8

40. log 4 兹7

1 logb x 2

63. Evaluate the following without using a calculator. log3 5 log5 7 log7 9 64. Evaluate the following without using a calculator. log5 20 log20 60 log60 100 log100 125 65. Which is larger, 500501 or 506500? These numbers are too large for most calculators to handle. (They each have 1353 digits!) (Hint: Compare the logarithms of each number.)

7.3

Properties of Logarithms and Logarithmic Scales

491

1 1 66. Which number is smaller, 300 or ? See hint in 50 151233 Exercise 65.

where I0 is the intensity of sound that is barely audible to the human ear. Use the decibel level formula to work exercises 73-76.

67. ANIMATED MAPS A software company that creates interactive maps for websites has designed an animated zooming feature such that when a user selects the zoom-in option, the map appears to expand on a location. This is accomplished by displaying several intermediate maps to give the illusion of motion. The company has determined that zooming in on a location is more informative and pleasing to observe when the scale of each step of the animation is determined using the equation

73. Find the decibel level for the following sounds. Round to the nearest tenth of a decibel.

n

Sn 苷 S0 10 N

Sound

共log Sf log S0兲

where Sn represents the scale of the current step n (n 苷 0 corresponds to the initial scale), S0 is the starting scale of the map, Sf is the final scale, and N is the number of steps in the animation following the initial scale. (If the initial scale of the map is 1⬊200, then S0 苷 200.) Determine the scales to be used at each intermediate step if a map is to start with a scale of 1⬊1,000,000 and proceed through five intermediate steps to end with a scale of 1⬊500,000.

Intensity

a. Automobile traffic

I 苷 1.58 10 8 I0

b. Quiet conversation

I 苷 10,800 I0

c. Fender guitar

I 苷 3.16 10 11 I0

d. Jet engine

I 苷 1.58 10 15 I0

74. COMPARISON OF SOUND INTENSITIES A team in Arizona installed a 48,000-watt sound system in a Ford Bronco that it claims can output 175-decibel sound. The human pain threshold for sound is 125 decibels. How many times as great is the intensity of the sound from the Bronco than the human pain threshold? 75. COMPARISON OF SOUND INTENSITIES How many times as great is the intensity of a sound that measures 120 decibels than a sound that measures 110 decibels?

68. ANIMATED MAPS Use the equation in Exercise 67 to determine the scales for each stage of an animated map zoom that goes from a scale of 1⬊250,000 to a scale of 1⬊100,000 in four steps (following the initial scale).

76. DECIBEL LEVEL If the intensity of a sound is doubled, what is the increase in the decibel level? (Hint: Find dB共2I兲 dB共I兲.)

69. pH Milk of magnesia has a hydronium-ion concentration of about 3.97 10 11 mole per liter. Determine the pH of milk of magnesia and state whether milk of magnesia is an acid or a base.

78.

EARTHQUAKE MAGNITUDE The Colombia earthquake of 1906 had an intensity of I 苷 398,107,000I0. What did it measure on the Richter scale?

70. pH Vinegar has a hydronium-ion concentration of 1.26 10 3 mole per liter. Determine the pH of vinegar and state whether vinegar is an acid or a base.

79.

EARTHQUAKE INTENSITY The Coalinga, California, earthquake of 1983 had a Richter scale magnitude of 6.5. Find the intensity of this earthquake.

71. HYDRONIUM-ION CONCENTRATION A morphine solution has a pH of 9.5. Determine the hydronium-ion concentration of the morphine solution.

80.

EARTHQUAKE INTENSITY The earthquake that occurred just south of Concepción, Chile, in 1960 had a Richter scale magnitude of 9.5. Find the intensity of this earthquake.

72. HYDRONIUM-ION CONCENTRATION A rainstorm in New York City produced rainwater with a pH of 5.6. Determine the hydronium-ion concentration of the rainwater. DECIBEL LEVEL The range of sound intensities that the human ear can detect is so large that a special decibel scale (named after Alexander Graham Bell) is used to measure and compare sound intensities. The decibel level dB of a sound is given by

dB共I 兲 10 log

冉冊 I I0

77. EARTHQUAKE MAGNITUDE What is the Richter scale magnitude of an earthquake with an intensity of I 苷 100,000I0?

81. COMPARISON OF EARTHQUAKES Compare the intensity of an earthquake that measures 5.0 on the Richter scale to the intensity of an earthquake that measures 3.0 on the Richter scale by finding the ratio of the larger intensity to the smaller intensity. 82.

COMPARISON OF EARTHQUAKES How many times as great was the intensity of the 1960 earthquake in Chile, which measured 9.5 on the Richter scale, than the San Francisco earthquake of 1906, which measured 8.3 on the Richter scale?

492 83.

84.

Chapter 7

Exponential and Logarithmic Functions

COMPARISON OF EARTHQUAKES On March 2, 1933, an earthquake of magnitude 8.9 on the Richter scale struck Japan. In October 1989, an earthquake of magnitude 7.1 on the Richter scale struck San Francisco, California. Compare the intensity of the larger earthquake to the intensity of the smaller earthquake by finding the ratio of the larger intensity to the smaller intensity. COMPARISON OF EARTHQUAKES An earthquake that occurred in China in 1978 measured 8.2 on the Richter scale. In 1988, an earthquake in California measured 6.9 on the Richter scale. Compare the intensity of the larger earthquake to the intensity of the smaller earthquake by finding the ratio of the larger intensity to the smaller intensity.

86. EARTHQUAKE MAGNITUDE Find the Richter scale magnitude of the earthquake that produced the seismogram in the following figure. s-wave

t = 17 seconds

87. Prove the quotient property of logarithms: logb

85. EARTHQUAKE MAGNITUDE Find the Richter scale magnitude of the earthquake that produced the seismogram in the following figure. s-wave p-wave

A = 18 mm

A = 26 mm

p-wave

M 苷 logb M logb N N

(Hint: See the proof of the product property of logarithms on page 481.) 88. Prove the power property of logarithms: logb 共M p 兲苷 p logb M See the hint given in exercise 87.

t = 31 seconds

Connecting Concepts 30

200 100 60 40

20 5

100

40

6

50

30

5

20

20 10 6 4

4 3 2 1

2

Amplitude (mm)

300

50

Magnitude

400

10 Amplitude = 23 mm

0 10 20 S − P = 24

S−P time (s)

500

20

S

P

Distance (km)

89. NOMOGRAMS AND LOGARITHMIC SCALES A nomogram is a diagram used to determine a numerical result by drawing a line across numerical scales. The following nomogram, used by Richter, determines the magnitude of an earthquake from its seismogram. To use the nomogram, mark the amplitude of a seismogram on the amplitude scale and mark the time between the s-wave and the p-wave on the S-P scale. Draw a line between these marks. The Richter scale magnitude of the earthquake that produced the seismogram is shown by the intersection of the line and the center scale. The nomogram on the right shows that an earthquake with a seismogram amplitude of 23 millimeters and an S-P time of 24 seconds has a Richter scale magnitude of about 5. The amplitude and the S-P time are shown on logarithmic scales. On the amplitude scale, the distance from 1 to 10 is the same as the distance from 10 to 100, because log 100 log 10 苷 log 10 log 1.

10 5 2 1 0.5 0.2 0.1

0

0

Richter’s earthquake nomogram

7.3

Properties of Logarithms and Logarithmic Scales

493

b. amplitude of 1 millimeter and S-P time of 30 seconds

Use the nomogram on page 492 to determine the Richter scale magnitude of an earthquake with a seismogram that has

c.

a. amplitude of 50 millimeters and S-P time of 40 seconds

How do the results in a. and b. compare with the Richter scale magnitude produced by using the amplitude-time-difference formula on page 487?

Projects 1.

LOGARITHMIC SCALES Sometimes logarithmic scales are used to better view a collection of data that span a wide range of values. For instance, consider the table below, which lists the approximate masses of various marine creatures in grams. Next we have attempted to plot the masses on a number line. Animal

Planet

Mercury Venus

Mass (g)

Rotifer

0.000000006

Dwarf goby

0.30

Lobster

15,900

Leatherback turtle

851,000

Giant squid

1,820,000

Whale shark

4,700,000

Blue whale

Distance (million km)

58 108

Earth

150

Mars

228

Jupiter

778

Saturn

1427

Uranus

2871

Neptune

4497

a. Draw a number line with an appropriate scale to plot the distances.

120,000,000 b. Draw a second number line, this time plotting the logarithm (base 10) of each distance.

0

20

40

60

80

100

120

c. Which number line do you find more helpful to compare the different distances?

Mass (in millions of grams)

As you can see, we had to use such a large span of numbers that the data for most of the animals are bunched up at the left. Visually, this number line isn’t very helpful for comparisons. a. Make a new number line, this time plotting the logarithm (base 10) of each of the masses. b. Which number line is more helpful to compare the masses of the different animals? c. If the data points for two animals on the logarithmic number line are 1 unit apart, how do the animals’ masses compare? What if the points are 2 units apart? 2.

LOGARITHMIC SCALES The distances of the planets in our solar system from the sun are given in the table in the next column.

d. If two distances are 3 units apart on the logarithmic number line, how do the distances of the corresponding planets compare? 3.

BIOLOGIC DIVERSITY To discuss the variety of species that live in a certain environment, a biologist needs a precise definition of diversity. Let p1 , p2 , . . . , pn be the proportions of n species that live in an environment. The biologic diversity D of this system is D 苷共 p1 log 2 p1 p2 log 2 p2 pn log 2 pn 兲 Suppose that an ecosystem has exactly five different varieties of grass: rye (R), bermuda (B), blue (L), fescue (F), and St. Augustine (A).

494

Chapter 7

Exponential and Logarithmic Functions environment and the proportions are as shown in Table 3. Calculate the diversity of this system. (Note: Although the equation is not technically correct, for purposes of the diversity definition, we may say that 0 log 2 0 苷 0. By using very small values of pi , we can demonstrate that this definition makes sense.) Does this system have more or less diversity than the system given in Table 2?

a. Calculate the diversity of this ecosystem if the proportions of these grasses are as shown in Table 1. Round to the nearest hundredth. Table 1 R

B

L

F

A

1 5

1 5

1 5

1 5

1 5

Table 3 b. Because bermuda and St. Augustine are virulent grasses, after a time the proportions will be as shown in Table 2. Calculate the diversity of this system. Does this system have more or less diversity than the system given in Table 1? B

L

F

A

1 8

3 8

1 16

1 8

5 16

■

■

Solve Exponential Equations Solve Logarithmic Equations

L

F

A

0

1 4

0

0

3 4

Table 4

c. After an even longer time period, the bermuda and St. Augustine grasses completely overrun the

Section 7.4

B

d. Finally, the St. Augustine grasses overrun the bermuda grasses and the proportions are as shown in Table 4. Calculate the diversity of this system. Write a sentence that explains the meaning of the value you obtained.

Table 2 R

R

R

B

L

F

A

0

0

0

0

1

Exponential and Logarithmic Equations P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A39.

PS1. Use the definition of a logarithm to write the exponential equation 36 苷 729 in logarithmic form. [7.2] PS2. Use the definition of a logarithm to write the logarithmic equation log5 625 苷 4 in exponential form. [7.2] PS3. Use the definition of a logarithm to write the exponential equation ax2 苷 b in logarithmic form. [7.2] PS4. Solve for x: 4a 苷 7bx 2cx. [1.1] PS5. Solve for x: 165 苷 PS6. Solve for x: A 苷

300 . [1.1] 1 12x

100 x . [1.1] 100 x

Solve Exponential Equations If a variable appears in the exponent of a term of an equation, such as in 2x1 苷 32, then the equation is called an exponential equation. Example 1 uses the following Equality of Exponents Theorem to solve 2x1 苷 32.

7.4

495

Exponential and Logarithmic Equations

Equality of Exponents Theorem If bx 苷 b y, then x 苷 y, provided b 0 and b 苷 1.

EXAMPLE 1

»

Solve an Exponential Equation

Use the Equality of Exponents Theorem to solve 2x1 苷 32.

Solution 2x1 苷 32 2x1 苷 25 x1苷5 x苷4 Check: Let x 苷 4. Then 2

x1

• Write each side as a power of 2. • Equate the exponents. • Solve for x.

苷2 苷 25 苷 32

41

»

Try Exercise 2, page 501

A graphing utility can also be used to find the solutions of an equation of the form f共x兲苷 g共x兲. Either of the following two methods can be employed. Using a Graphing Utility to Find the Solutions of f(x) g(x) Intersection Method Graph y1 苷 f共x兲 and y2 苷 g共x兲 on the same screen. The solutions of f共x兲苷 g共x兲 are the x-coordinates of the points of intersection of the graphs. Intercept Method The solutions of f共x兲苷 g共x兲 are the x-coordinates of the x-intercepts of the graph of y 苷 f共x兲 g共x兲. Figures 7.24 and 7.25 illustrate the graphical methods for solving 2x1 苷 32. 8

60

y1 = 2x + 1− 32

y2 = 32

0

Intersection X=4

x-intercept 9.4

Y=32

9.4

0

y1 = 2x + 1

Zero X=4

Y=0

−8

−15

Intersection method

Figure 7.24

Intercept method

Figure 7.25

In Example 1 we were able to write both sides of the equation as a power of the same base. If you find it difficult to write both sides of an exponential equation in terms of the same base, then try the procedure of taking the logarithm of each side of the equation. This procedure is used in Example 2.

496

Chapter 7

EXAMPLE 2

Solve: 5x 苷 40

»

Exponential and Logarithmic Functions

Solve an Exponential Equation

ALGEBRAIC SOLUTION 5x 苷 40 log共5x兲苷 log 40 x log 5 苷 log 40 log 40 x苷 log 5 x ⬇ 2.3

VISUALIZE THE SOLUTION

• Take the logarithm of each side. • Power property

Intersection Method The solution of 5x 苷 40 is the x-coordinate of the point of intersection of y 苷 5x and y 苷 40.

• Exact solution

60

• Decimal approximation

To the nearest tenth, the solution is 2.3.

y = 40 y = 5x 0

Intersection X=2.2920297 Y=40

4.7

−15

»

Try Exercise 10, page 501

An alternative approach to solving the equation in Example 2 is to rewrite the exponential equation in logarithmic form: 5x 苷 40 is equivalent to the logarithmic equation log 5 40 苷 x. Using the change-of-base formula, we find that log 40 x 苷 log 5 40 苷 . In the following example, however, we must take logalog 5 rithms of both sides to reach a solution. EXAMPLE 3

»

Solve an Exponential Equation

Solve: 32x1 苷 5x2

ALGEBRAIC SOLUTION 32x1 苷 5x2 ln 32x1 苷 ln 5x2 共2x 1兲 ln 3 苷共x 2兲 ln 5 2x ln 3 ln 3 苷 x ln 5 2 ln 5 2x ln 3 x ln 5 苷 2 ln 5 ln 3 x共2 ln 3 ln 5兲苷 2 ln 5 ln 3 2 ln 5 ln 3 x苷 2 ln 3 ln 5 x ⬇ 7.3 To the nearest tenth, the solution is 7.3.

»

Try Exercise 18, page 501

VISUALIZE THE SOLUTION

• Take the natural logarithm of each side.

Intercept Method The solution of 32x1 苷 5x2 is the x-coordinate of the x-intercept of y 苷 32x1 5x2.

• Power property

400,000

• Distributive property y = 3 2x − 1− 5 x + 2

• Solve for x. • Factor. • Exact solution • Decimal approximation

−6

12

Zero X=7.3453319 Y=0 −400,000

7.4

497

Exponential and Logarithmic Equations

In Example 4 we solve an exponential equation that has two solutions. EXAMPLE 4

Solve:

»

Solve an Exponential Equation Involving bx bx

2x 2x 苷3 2

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

Multiplying each side by 2 produces

Intersection Method The solutions 2x 2x 苷 3 are the of 2 x-coordinates of the points of 2x 2x intersection of y 苷 and 2 y 苷 3.

x

2 2 苷6 22x 20 苷 6共2x兲 x

• Multiply each side by 2x to clear negative exponents.

共2x 兲2 6共2x 兲 1 苷 0 共u兲2 6共u兲 1 苷 0

• Write in quadratic form. • Substitute u for 2x.

By the quadratic formula,

5

6 兹36 4 6 4兹2 苷苷 3 2兹2 2 2 2x 苷 3 2兹2 • Replace u with 2x. • Take the common log 2x 苷 log共 3 2兹2 兲

y=3

u苷

x log 2 苷 log共 3 2兹2 兲 log共 3 2兹2 兲 x苷 ⬇ 2.54 log 2

y=

(−2.54, 3) −4

logarithm of each side.

(2.54, 3) 2 x + 2 −x 2 4

−1

• Power property • Solve for x.

The approximate solutions are 2.54 and 2.54.

»

Try Exercise 42, page 501

Solve Logarithmic Equations Equations that involve logarithms are called logarithmic equations. The properties of logarithms, along with the definition of a logarithm, are often used to find the solutions of a logarithmic equation. EXAMPLE 5

»

Solve a Logarithmic Equation

Solve: log共3x 5兲苷 2

Solution log共3x 5兲苷 2 3x 5 苷 10 2 3x 苷 105 x 苷 35 Check: log关3共35兲 5兴苷 log 100 苷 2

»

Try Exercise 22, page 501

• Definition of a logarithm • Solve for x.

498

Chapter 7

Exponential and Logarithmic Functions EXAMPLE 6

»

Solve a Logarithmic Equation

Solve: log 2x log共x 3兲苷 1

Solution log 2x log共x 3兲苷 1 log

2x 苷1 x3

• Quotient property

2x 苷 10 1 x3

• Definition of a logarithm

2x 苷 10x 30 8x 苷 30 x苷

• Multiply each side by x 3. • Solve for x.

15 4

Check the solution by substituting

15 into the original equation. 4

»

Try Exercise 26, page 501

In Example 7 we make use of the one-to-one property of logarithms to find the solution of a logarithmic equation.

EXAMPLE 7

»

Solve a Logarithmic Equation

Solve: ln共3x 8兲苷 ln共2x 2兲 ln共x 2兲

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

ln共3x 8兲苷 ln共2x 2兲 ln共x 2兲 ln共3x 8兲苷 ln关共2x 2兲共x 2兲兴 ln共3x 8兲苷 ln共2x 2x 4兲 3x 8 苷 2x 2 2x 4

The graph of • Product property

2

0 苷 2x 2 5x 12 0 苷共2x 3兲共x 4兲 3 x苷 2

or

x苷4

• One-to-one property of logarithms • Subtract 3x 8 from each side.

Try Exercise 36, page 501

has only one x-intercept. Thus there is only one real solution. y

2

• Factor. • Solve for x.

A check will show that 4 is a solution, but

»

y 苷 ln共3x 8兲 ln共2x 2兲 ln共x 2兲

3 is not a solution. 2

x=4 2

4

x

7.4

QUESTION

Why does x 苷

EXAMPLE 8

499

Exponential and Logarithmic Equations

»

3 not check in Example 7? 2

Velocity of a Sky Diver Experiencing Air Resistance

During the free-fall portion of a jump, the time t in seconds required for a sky diver to reach a velocity v in feet per second is given by t苷

take note

Determine the velocity of the diver after 5 seconds.

b.

The graph of t has a vertical asymptote at v 苷 175. Explain the meaning of the vertical asymptote in the context of this example.

Solution Substitute 5 for t and solve for v.

for a sky diver to reach a given velocity (in feet per second) is t苷

required to reach a given velocity

5苷

175 v ln 1 32 175

冊冊

v 32 苷 ln 1 35 175

e32/35 苷 1 e32/35 1 苷

during the free-fall of a sky diver

v 175

v 175

v 苷 175共1 e32/35兲

who is experiencing air resistance.

冊冊

175 v ln 1 32 175

32 v 5 苷 ln 1 175 175

v . The function in Example 8 32

is a more realistic model of the time

冉冉

t苷

冉冊冉冉

then the time in seconds required

冊

a.

a.

If air resistance is not considered,

冉

175 v ln 1 , 0 v 175 32 175

• Replace t with 5. 32

• Multiply each side by . 175

• Simplify.

• Write in exponential form.

• Subtract 1 from each side. • Multiply each side by 175.

v ⬇ 104.86 Continued 䉴

ANSWER

冉冊

冉冊

3 7 7 , the original equation becomes ln 苷 ln共1兲 ln . 2 2 2 This cannot be true, because the function f 共x兲苷 ln x is not defined for negative values of x. If x 苷

Chapter 7

Exponential and Logarithmic Functions After 5 seconds the velocity of the sky diver will be about 104.9 feet per second. See Figure 7.26.

t v = 175

20 Time (in seconds)

500

15 10 5 104.9 50

100

150

v

Velocity (in feet per second)

t = − 175 ln(1 − v ) 32 175

Figure 7.26

The vertical asymptote v 苷 175 indicates that the velocity of the sky diver approaches, but never reaches or exceeds, 175 feet per second. In Figure 7.26, note that as v l 175 from the left, t l .

b.

»

Try Exercise 74, page 504

Topics for Discussion 1. Discuss how to solve the equation a 苷 log b x for x. 2. What is the domain of y 苷 log4共2x 5兲? Explain why this means that the equation log4共x 3兲苷 log4共2x 5兲 has no real number solution. 3. 8 is not a solution of the equation log 2 x log 2共x 6兲苷 4. Discuss at which step in the following solution the extraneous solution 8 was introduced. log 2 x log 2共x 6兲苷 4 log 2 x共x 6兲苷 4 x共x 6兲苷 24 x 2 6x 苷 16 x 2 6x 16 苷 0 共x 8兲共x 2兲苷 0 x 苷 8 or x苷2

7.4

Exponential and Logarithmic Equations

501

Exercise Set 7.4 In Exercises 1 to 48, use algebraic procedures to find the exact solution(s) of the equation.

1. 2x 苷 64

2. 3x 苷 243

1 3. 49 苷 343

1 4. 9 苷 243

x

5. 25x3 苷

7.

冉冊 2 5

x

苷

6. 34x7 苷

8 125

8.

冉冊 2 5

x

25 4

9. 5 苷 70

10. 6 苷 50

11. 3x 苷 120

12. 7x 苷 63

13. 10 2x3 苷 315

14. 10 6x 苷 550

15. ex 苷 10

16. ex1 苷 20

17. 21x 苷 3x1

18. 3x2 苷 42x1

19. 22x3 苷 5x1

20. 53x 苷 3x4

21. log共4x 18兲苷 1

22. log共x 19兲苷 2

x

冊

1 ln 2 2

38. log4 x log4(x 2) 苷 log4 15

1 9

苷

冉

5 1 ln 2x 2 2

37. log7(5x) log7 3 苷 log7(2x 1)

x

1 8

36. ln x 苷

x

2

39. eln共x1兲苷 4

40. 10 log共2x7兲苷 8

41.

10 x 10 x 苷 20 2

42.

10 x 10 x 苷8 2

43.

10 x 10 x 苷5 10 x 10 x

44.

1 10 x 10 x 苷 10 x 10 x 2

45.

e x ex 苷 15 2

46.

e x ex 苷 15 2

47.

1 苷4 e ex

48.

e x ex 苷3 e x ex

x

In Exercises 49 to 58, use a graphing utility to approximate the solution(s) of the equation to the nearest hundredth.

23. ln共x 2 12兲苷 ln x

49. 2x3 苷 x 1

50. 3x2 苷 2x 1

24. log共2x 2 3x兲苷 log共10x 30兲

51. e 32x 2x 苷 1

52. 2e x2 3x 苷 2

53. 3 log 2共x 1兲苷 x 3

54. 2 log 3共2 3x兲苷 2x 1

25. log 2 x log 2共x 4兲苷 2 26. log 3 x log 3共x 6兲苷 3 27. log共5x 1兲苷 2 log共x 2兲 28. 1 log共3x 1兲苷 log共2x 1兲

30. log共4 x兲苷 log共x 8兲 log共2x 13兲

33. log共log x兲苷 1 35. ln共e3x兲苷 6

1 2

57. 2x1 苷 x 2 1

1 x 苷 3 2

56. 2 ln共3 x兲 3x 苷 4 58. ln x 苷 x 2 4

59. POPULATION GROWTH The population P of a city grows exponentially according to the function

29. ln共1 x兲 ln共3 x兲苷 ln 8

31. log 兹x 3 17 苷

55. ln共2x 4兲

32. log共x 3兲苷共log x兲2 34. ln共ln x兲苷 2

P共t兲苷 8500共1.1兲t,

0t8

where t is measured in years. a. Find the population at time t 苷 0 and also at time t 苷 2. b. When, to the nearest year, will the population reach 15,000?

502

Chapter 7

Exponential and Logarithmic Functions

60. PHYSICAL FITNESS After a race, a runner’s pulse rate R in beats per minute decreases according to the function R共t兲苷 145e0.092t,

0 t 15

64. A haddock has a length of 21 centimeters. Use Bertalanffy’s equation to predict, to the nearest tenth of a year, the age of the haddock. Assume m 苷 94 centimeters, L 0 苷 0.6 centimeters, and r 苷 0.21.

where t is measured in minutes. a. Find the runner’s pulse rate at the end of the race and also 1 minute after the end of the race. b. How long, to the nearest minute, after the end of the race will the runner’s pulse rate be 80 beats per minute? 61. RATE OF COOLING A can of soda at 79°F is placed in a refrigerator that maintains a constant temperature of 36°F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by T共t兲苷 36 43e0.058t a. Find the temperature, to the nearest degree, of the soda 10 minutes after it is placed in the refrigerator. b. When, to the nearest minute, will the temperature of the soda be 45°F? 62. MEDICINE During surgery, a patient’s circulatory system requires at least 50 milligrams of an anesthetic. The amount of anesthetic present t hours after 80 milligrams of anesthetic is administered is given by T共t兲苷 80共0.727兲t a. How much, to the nearest milligram, of the anesthetic is present in the patient’s circulatory system 30 minutes after the anesthetic is administered? b. How long, to the nearest minute, can the operation last if the patient does not receive additional anesthetic? BERTALANFFY’S EQUATION In 1938, the biologist Ludwig von Bertalanffy developed the equation L m (m L0)e rx which models the length L, in centimeters, of a fish as it grows under optimal conditions for a period of x years. In Bertalanffy’s equation, m represents the maximum length, in centimeters, the fish is expected to attain, L0 is the length, in centimeters, of the fish at birth, and r is a constant related to the growth rate of the fish species. Use Bertalanffy’s equation to predict the age of the fish described in Exercises 63 and 64.

63. A barracuda has a length of 114 centimeters. Use Bertalanffy’s equation to predict, to the nearest tenth of a year, the age of the barracuda. Assume m 苷 198 centimeters, L 0 苷 0.9 centimeter, and r 苷 0.23.

65. TYPING SPEED The following function models the average typing speed S, in words per minute, for a student who has been typing for t months. S(t) 苷 5 29 ln(t 1), 0 t 9 Use S to determine how long it takes the student to achieve an average typing speed of 65 words per minute. Round to the nearest tenth of a month. 66. WALKING SPEED An approximate relation between the average pedestrian walking speed s, in miles per hour, and the population x, in thousands, of a city is given by the formula s(x) 苷 0.37 ln x 0.05 Use s to estimate the population of a city for which the average pedestrian walking speed is 2.9 miles per hour. Round to the nearest hundred thousand. 67. DRAG RACING The quadratic function s1(x) 苷 2.25x 2 56.26x 0.28,

0 x 10

models the speed of a dragster from the start of a race until the dragster crosses the finish line 10 seconds later. This is the acceleration phase of the race. The exponential function s2(x) 苷 8320(0.73)x,

10 x 20

models the speed of the dragster during the 10-second period immediately following the time when the dragster crosses the finish line. This is the deceleration period. How long after the start of the race did the dragster attain a speed of 275 miles per hour? Round to the nearest hundredth of a second. 68. EIFFEL TOWER The functions h1(x) 苷 363.4 88.4 ln x,

16.47 x 61.0

h2(x) 苷 568.2 161.5 ln x,

6.1 x 16.47

and

approximate the height, in meters, of the Eiffel Tower x meters to the right of the center line, shown by the y-axis in the following figure (on page 503).

7.4 y Third stage 276.13 m

Exponential and Logarithmic Equations 70.

300

503

PSYCHOLOGY An industrial psychologist has determined that the average percent score for an employee on a test of the employee’s knowledge of the company’s product is given by P苷

h2

100 1 40e0.1t

2000 00

where t is the number of weeks on the job and P is the percent score. a. Use a graphing utility to graph the equation for t 0.

Second stage 115.73 m

b. Use the graph to estimate (to the nearest week) the expected number of weeks of employment that are necessary for an employee to earn a 70% score on the test.

100 00 h1

First stage 57.63 m

c. Determine the horizontal asymptote of the graph. −50

50

x

The graph of h1 models the shape of the tower from ground level up to the second stage in the figure, and the graph of h2 models the shape of the tower from the second stage up to the third stage. Determine the horizontal distance across the Eiffel Tower, rounded to the nearest tenth of a meter, at a height of

d.

71.

Write a sentence that explains the meaning of the horizontal asymptote.

ECOLOGY A herd of bison was placed in a wildlife preserve that can support a maximum of 1000 bison. A population model for the bison is given by B苷

1000 1 30e0.127t

a. 50 meters where B is the number of bison in the preserve and t is time in years, with the year 1999 represented by t 苷 0.

b. 125 meters 69.

a. Use a graphing utility to graph the equation for t 0.

PSYCHOLOGY Industrial psychologists study employee training programs to assess the effectiveness of the instruction. In one study, the percent score P on a test for a person who had completed t hours of training was given by P苷

b. Use the graph to estimate (to the nearest year) the number of years before the bison population reaches 500. c. Determine the horizontal asymptote of the graph.

100 1 30e0.088t

d.

a. Use a graphing utility to graph the equation for t 0. b. Use the graph to estimate (to the nearest hour) the number of hours of training necessary to achieve a 70% score on the test. c. From the graph, determine the horizontal asymptote. d.

Write a sentence that explains the meaning of the horizontal asymptote.

72.

Write a sentence that explains the meaning of the horizontal asymptote. POPULATION GROWTH A yeast culture grows according to the equation Y苷

50,000 1 250e0.305t

where Y is the number of yeast and t is time in hours. a. Use a graphing utility to graph the equation for t 0.

504

Chapter 7

Exponential and Logarithmic Functions b. Determine the horizontal asymptote for the graph of this function.

b. Use the graph to estimate (to the nearest hour) the number of hours before the yeast population reaches 35,000.

c. c. From the graph, estimate the horizontal asymptote. d.

Write a sentence that explains the meaning of the horizontal asymptote. CONSUMPTION OF NATURAL RESOURCES A model for how long our coal resources will last is given by

73.

s苷

冉

冊

100 2 e0.32t e0.32t ln 32 2

a. Use a graphing utility to graph this equation for t 0.

where r is the percent increase in consumption from current levels of use and T is the time (in years) before the resources are depleted.

b. How long does it take for the object to fall 100 feet? Round to the nearest tenth of a second.

b. If our consumption of coal increases by 3% per year, in how many years will we deplete our coal resources? c. What percent increase in consumption of coal will deplete the resources in 100 years? Round to the nearest tenth of a percent. EFFECTS OF AIR RESISTANCE ON VELOCITY If we assume that air resistance is proportional to the square of the velocity, then the time t in seconds required for an object to reach a velocity v in feet per second is given by t苷

24 v 9 ln , 0 v 24 24 24 v

a. Determine the velocity, to the nearest hundredth of a foot per second, of the object after 1.5 seconds. b. Determine the vertical asymptote for the graph of this function. c.

75.

EFFECTS OF AIR RESISTANCE ON DISTANCE The distance s, in feet, that the object in Exercise 75 will fall in t seconds is given by

ln共300r 1兲 T苷 ln共r 1兲

a. Use a graphing utility to graph this equation.

74.

76.

Write a sentence that explains the meaning of the horizontal asymptote in the context of this application.

77. RETIREMENT PLANNING The retirement account for a graphic designer contains $250,000 on January 1, 2006, and earns interest at a rate of 0.5% per month. On February 1, 2006, the designer withdraws $2000 and plans to continue these withdrawals as retirement income each month. The value V of the account after x months is V 苷 400,000 150,000共1.005兲x If the designer wishes to leave $100,000 to a scholarship foundation, what is the maximum number of withdrawals the designer can make from this account and still have $100,000 to donate? 78. HANGING CABLE The height h, in feet, of any point P on the cable shown is given by h共x兲苷 10共e x/20 e x/20兲,

where 兩x兩 is the horizontal distance in feet between P and the y-axis.

y

Write a sentence that explains the meaning of the vertical asymptote in the context of this application.

TERMINAL VELOCITY WITH AIR RESISTANCE The velocity v of an object t seconds after it has been dropped from a height above the surface of the earth is given by the equation v 苷 32t feet per second, assuming no air resistance. If we assume that air resistance is proportional to the square of the velocity, then the velocity after t seconds is given by

冉

v 苷 100

冊

e0.64t 1 e0.64t 1

a. In how many seconds will the velocity be 50 feet per second?

15 x 15

P

−15

15

x

a. What is the lowest height of the cable? b. What is the height of the cable 10 feet to the right of the y-axis? Round to the nearest tenth of a foot. c. How far to the right of the y-axis is the cable 24 feet in height? Round to the nearest tenth of a foot.

7.4

505

Exponential and Logarithmic Equations

Connecting Concepts 79. The following argument seems to indicate that 0.125 0.25. Find the first incorrect statement in the argument. 32 3共log 0.5兲 2共log 0.5兲 log 0.53 log 0.52 0.53 0.52

81. A common mistake that students make is to write log共x y兲 as log x log y. For what values of x and y does log共x y兲苷 log x log y? (Hint: Solve for x in terms of y.) 82. Let f 共x兲苷 2 ln x and g共x兲苷 ln x 2. Does f 共x兲苷 g共x兲 for all real numbers x? 83.

0.125 0.25 80. The following argument seems to indicate that 4 苷 6. Find the first incorrect statement in the argument. 4 苷 log 2 16

Explain why the functions F共x兲苷 1.4x and G共x兲苷 e0.336x represent essentially the same function.

84. Find the constant k that will make f 共t兲苷 2.2t and g共t兲苷 ekt represent essentially the same function.

4 苷 log 2共8 8兲 4 苷 log 2 8 log 2 8 4苷33 4苷6

Projects 1.

NAVIGATING The pilot of a boat is trying to cross a river to a point O two miles due west of the boat’s starting position by always pointing the nose of the boat toward O. Suppose the speed of the current is w miles per hour and the speed of the boat is v miles per hour. If point O is the origin and the boat’s starting position is 共2, 0兲 (see the diagram at the right), then the equation of the boat’s path is given by y苷

冉冊 x 2

1共w/v兲

冉冊 x 2

1共w/v兲

ing the nose of the boat toward point O? If not, where will the pilot arrive? Explain. c. If the speed of the current is less than the speed of the boat, can the pilot reach point O by always pointing the nose of the boat toward point O? If not, where will the pilot arrive? Explain. y N

a. If the speed of the current and the speed of the boat are the same, can the pilot reach point O by always having the nose of the boat pointed toward O? If not, at what point will the pilot arrive? Explain your answer.

v O

b. If the speed of the current is greater than the speed of the boat, can the pilot reach point O by always point-

w Current

(2, 0)

x

506

Chapter 7

Section 7.5 ■

■ ■

■

Exponential Growth and Decay Carbon Dating Compound Interest Formulas Restricted Growth Models

Exponential and Logarithmic Functions

Exponential Growth and Decay P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A40.

冉冊冉冊

PS1. Evaluate A 苷 1000 1 PS2. Evaluate A 苷 600 1

0.1 12

0.04 4

12t

4t

for t 苷 2. Round to the nearest hundredth. [7.1]

for t 苷 8. Round to the nearest hundredth. [7.1]

PS3. Solve 0.5 苷 e14k for k. Round to the nearest ten-thousandth. [7.4] PS4. Solve 0.85 苷 0.5t/5730 for t. Round to the nearest ten. [7.4] PS5. Solve 6 苷

70 for k. Round to the nearest thousandth. [7.4] 5 9ek12

PS6. Solve 2,000,000 苷

3n1 3 for n. Round to the nearest tenth. [7.4] 2

Exponential Growth and Decay In many applications, a quantity changes at a rate proportional to the amount present. In these applications, the amount present at time t is given by a special function called an exponential growth function or an exponential decay function. Definition of Exponential Growth and Decay Functions If a quantity N increases or decreases at a rate proportional to the amount present at time t, then the quantity can be modeled by N共t兲苷 N0 e kt where N0 is the value of N at time t 苷 0, and k is a constant called the growth rate constant. ■

If k is positive, N increases as t increases and N共t兲苷 N0 e kt is called an exponential growth function. See Figure 7.27.

■

If k is negative, N decreases as t increases and N共t兲苷 N0 e kt is called an exponential decay function. See Figure 7.28. N

N

N0

N0

N(t) = N0 e kt; k < 0, t ≥ 0

N(t) = N0 e kt; k > 0, t ≥ 0

t

t

Figure 7.27

Figure 7.28

Exponential growth function

Exponential decay function

7.5 QUESTION

Exponential Growth and Decay

507

Is N共t兲苷 1450e0.05t an exponential growth function or an exponential decay function?

In Example 1 we find an exponential growth function that models the population growth of a city. EXAMPLE 1

»

Find the Exponential Growth Function That Models Population Growth

a.

The population of a city is growing exponentially. The population of the city was 16,400 in 1995 and 20,200 in 2005. Find the exponential growth function that models the population growth of the city.

b.

Use the function from a. to predict, to the nearest 100, the population of the city in 2010.

Solution We need to determine N0 and k in N共t兲苷 N0 e kt. If we represent the year 1995 by t 苷 0, then our given data are N共0兲苷 16,400 and N共10兲苷 20,200. Because N0 is defined to be N共0兲, we know that N0 苷 16,400. To determine k, substitute t 苷 10 and N0 苷 16,400 into N共t兲苷 N0 e kt to produce

a.

N共10兲苷 16,400ek10

ln

20,200 苷 16,400e10k

• Substitute 20,200 for N(10).

20,200 苷 e10k 16,400

• Solve for e10k.

20,200 苷 10k 16,400

• Write in logarithmic form.

1 20,200 ln 苷k 10 16,400

• Solve for k.

0.0208 ⬇ k The exponential growth function is N共t兲 ⬇ 16,400e0.0208t. The year 1995 was represented by t 苷 0, so we will use t 苷 15 to represent the year 2010.

b.

N共t兲 ⬇ 16,400e0.0208t N共15兲 ⬇ 16,400e0.020815 ⬇ 22,400

(nearest 100)

The exponential growth function yields 22,400 as the approximate population of the city in 2010.

»

Try Exercise 6, page 516

ANSWER

Because the growth rate constant k 苷 0.05 is positive, the function is an exponential growth function.

508

Chapter 7

Exponential and Logarithmic Functions Many radioactive materials decrease in mass exponentially over time. This decrease, called radioactive decay, is measured in terms of half-life, which is defined as the time required for the disintegration of half the atoms in a sample of a radioactive substance. Table 7.11 shows the half-lives of selected radioactive isotopes. Table 7.11 Isotope

Half-Life

Carbon 共14C兲

5730 years

Radium 共226Ra兲

1660 years

Polonium 共 Po兲

138 days

210

Phosphorus 共 P兲

14 days

Polonium 共214Po兲

1兾10,000th of a second

32

EXAMPLE 2

»

Find an Exponential Decay Function

Find the exponential decay function for the amount of phosphorus (32P) that remains in a sample after t days.

Solution

take note

When t 苷 0, N共0兲苷 N0e k共0兲苷 N0. Thus N共0兲苷 N0. Also, because the phosphorus has a half-life of 14 days (from Table 7.11), N共14兲苷 0.5N0. To find k, substitute t 苷 14 into N共t兲苷 N0e kt and solve for k.

Because e0.0495 ⬇ 共0.5兲1/14, the decay function N共t兲苷 N0e0.0495t can also be written as N共t兲苷 N0共0.5兲t/14. In this form it is

N共14兲苷 N0 e k14

easy to see that if t is increased

0.5N0 苷 N0e 14k

by 14, then N will decrease by a

0.5 苷 e

factor of 0.5.

14k

ln 0.5 苷 14k 1 ln 0.5 苷 k 14

• Divide each side by N0. • Write in logarithmic form. • Solve for k.

0.0495 ⬇ k

P(t)

The exponential decay function is N共t兲 ⬇ N0e0.0495t.

N0 Amount of carbon-14

• Substitute 0.5N0 for N(14).

»

Try Exercise 8, page 516

0.75 N0

0.50 N0

Carbon Dating

0.25 N0

5730

17,190

28,650

Time (in years)

P(t) = 0.5t/5730

Figure 7.29

t

The bone tissue in all living animals contains both carbon-12, which is nonradioactive, and carbon-14, which is radioactive with a half-life of approximately 5730 years. See Figure 7.29. As long as the animal is alive, the ratio of carbon-14 to carbon-12 remains constant. When the animal dies 共t 苷 0兲, the carbon-14 begins to decay. Thus a bone that has a smaller ratio of carbon-14 to carbon-12 is older than a bone that has a larger ratio. The percent of carbon-14 present at time t is P共t兲苷 0.5t/5730

7.5

Math Matters The chemist Willard Frank Libby developed the carbon dating process in 1947. In 1960 he was awarded the Nobel Prize in chemistry for this achievement.

Exponential Growth and Decay

509

The process of using the percent of carbon-14 present at a given time to estimate the age of a bone is called carbon dating.

EXAMPLE 3

»

A Carbon Dating Application

Estimate the age of a bone if it now has 85% of the carbon-14 it had at time t 苷 0.

Solution Let t be the time at which P共t兲苷 0.85. 0.85 苷 0.5t/5730 ln 0.85 苷 ln 0.5t/5730 ln 0.85 苷

冉冊

5730

t ln 0.5 5730

ln 0.85 苷t ln 0.5 1340 ⬇ t

• Take the natural logarithm of each side. • Power property. • Solve for t.

The bone is about 1340 years old.

»

Try Exercise 12, page 516

Compound Interest Formulas Interest is money paid for the use of money. The interest I is called simple interest

if it is a fixed percent r, per time period t, of the amount of money invested. The amount of money invested is called the principal P. Simple interest is computed using the formula I 苷 Prt. For example, if $1000 is invested at 12% for 3 years, the simple interest is I 苷 Prt 苷 $1000共0.12兲共3兲苷 $360 The balance after t years is A 苷 P I 苷 P Prt. In the preceding example, the $1000 invested for 3 years produced $360 interest. Thus the balance after 3 years is $1000 $360 苷 $1360. In many financial transactions, interest is added to the principal at regular intervals so that interest is paid on interest as well as on the principal. Interest earned in this manner is called compound interest. For example, if $1000 is invested at 12% annual interest compounded annually for 3 years, then the total interest after 3 years is First-year interest Second-year interest Third-year interest

$1000共0.12兲苷 $120.00 $1120共0.12兲苷 $134.40 $1254.40共0.12兲 ⬇ $150.53 $404.93

• Total interest

This method of computing the balance can be tedious and time-consuming. A compound interest formula that can be used to determine the balance due after t years of compounding can be developed as follows.

510

Chapter 7

Table 7.12 Number of Years

Balance

3

A3 苷 P共1 r兲3

4

A4 苷 P共1 r兲4

. . .

. . .

t

At 苷 P共1 r兲t

Exponential and Logarithmic Functions Note that if P dollars is invested at an interest rate of r per year, then the balance after 1 year is A1 苷 P Pr 苷 P共1 r兲, where Pr represents the interest earned for the year. Observe that A1 is the product of the original principal P and 共1 r兲. If the amount A1 is reinvested for another year, then the balance after the second year is A2 苷共A1兲共1 r兲苷 P共1 r兲共1 r兲苷 P共1 r兲2 Successive reinvestments lead to the results shown in Table 7.12. The equation At 苷 P共1 r兲t is valid if r is the annual interest rate paid during each of the t years. If r is an annual interest rate and n is the number of compounding periods per year, then the interest rate each period is r兾n, and the number of compounding periods after t years is nt. Thus the compound interest formula is as follows: The Compound Interest Formula A principal P invested at an annual interest rate r, expressed as a decimal and compounded n times per year for t years, produces the balance

冉冊

A苷P 1

EXAMPLE 4

»

r n

nt

Solve a Compound Interest Application

Find the balance if $1000 is invested at an annual interest rate of 10% for 2 years compounded monthly

a.

b.

daily

Solution Because there are 12 months in a year, use n 苷 12.

a.

冉

A 苷 $1000 1

冊

122

0.1 12

⬇ $1000共1.008333333兲24 ⬇ $1220.39

Because there are 365 days in a year, use n 苷 365.

b.

冉

A 苷 $1000 1

冊

0.1 365

3652

⬇ $1000共1.000273973兲730 ⬇ $1221.37

»

Try Exercise 16, page 517

To compound continuously means to increase the number of compounding periods per year, n, without bound. 1 r To derive a continuous compounding interest formula, substitute for in m n the compound interest formula r nt A苷P 1 n

冉冊

to produce

7.5

冉

A苷P 1

冋冉

1 m

冊

nt

(1)

This substitution is motivated by the desire to express

1

1 m

冊册 m

511

Exponential Growth and Decay

r

冉冊 1

r n

n

as

, which approaches e r as m gets larger without bound.

1 r 苷 for n yields n 苷 mr, so the exponent nt can be m n written as mrt. Therefore Equation (1) can be expressed as Solving the equation

冉

A苷P 1

1 m

冊冋冉 mrt

苷P

1

1 m

冊册 m

rt

(2)

By the definition of e, we know that as m increases without bound,

冉

1

1 m

冊

m

approaches

e

Thus, using continuous compounding, Equation (2) simplifies to A 苷 Pert.

Continuous Compounding Interest Formula If an account with principal P and annual interest rate r is compounded continuously for t years, then the balance is A 苷 Pert.

EXAMPLE 5

»

Solve a Continuous Compound Interest Application

Find the balance after 4 years on $800 invested at an annual rate of 6% compounded continuously.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

Use the continuous compounding formula with P 苷 800, r 苷 0.06, and t 苷 4.

The following graph of A 苷 800e0.06t shows that the balance is about $1017.00 when t 苷 4.

A 苷 Pe rt 苷 800e0.06共4兲苷 800e0.24 ⬇ 800共1.27124915兲 ⬇ 1017.00

• Substitute given values.

1400

Y1=800e^(.06X)

• Simplify.

• Round to the nearest cent.

The balance after 4 years will be $1017.00.

9.4

0

X=4 −200

»

Try Exercise 18, page 517

Y=1016.9993

512

Chapter 7

Exponential and Logarithmic Functions You have probably heard it said that time is money. In fact, many investors ask the question “How long will it take to double my money?” The following example answers this question for two different investments. EXAMPLE 6

»

Double Your Money

Find the time required for money invested at an annual rate of 6% to double in value if the investment is compounded. semiannually

a.

Solution

continuously

b.

冉冊冉冊冉冊冉冊冉冊冉冊冉冊

r nt with r 苷 0.06, n 苷 2, and the balance A equal to n twice the principal 共A 苷 2P兲. Use A 苷 P 1

a.

2P 苷 P 1 2苷 1

0.06 2

0.06 2

ln 2 苷 ln 1

t苷

2t

0.06 2

ln 2 苷 2t ln 1 2t 苷

2t

2t

0.06 2

ln 2 0.06 ln 1 2 1 2

• Divide each side by P.

• Take the natural logarithm of each side.

• Apply the power property.

• Solve for t.

ln 2 0.06 ln 1 2

t ⬇ 11.72 If the investment is compounded semiannually, it will double in value in about 11.72 years. Use A 苷 Pe rt with r 苷 0.06 and A 苷 2P.

b.

2P 苷 Pe0.06t 2 苷 e0.06t ln 2 苷 0.06t t苷

ln 2 0.06

• Divide each side by P. • Write in logarithmic form. • Solve for t.

t ⬇ 11.55

»

If the investment is compounded continuously, it will double in value in about 11.55 years.

Try Exercise 22, page 517

7.5

513

Exponential Growth and Decay

Restricted Growth Models The exponential growth function N共t兲苷 N0 e kt is an unrestricted growth model that does not consider any limited resources that eventually will curb population growth. The logistic model is a restricted growth model that takes into consideration the effects of limited resources. The logistic model was developed by Pierre Verhulust in 1836. Definition of the Logistic Model (A Restricted Growth Model) The magnitude of a population at time t 0 is given by c P共t兲苷 1 aebt where c is the carrying capacity (the maximum population that can be supported by available resources as t l ) and b is a positive constant called the growth rate constant. The initial population is P0 苷 P共0兲. The constant a is related to the initial population P0 and the carrying capacity c by the formula c P0 a苷 P0

P(t) c

P0 t

P(t) =

c , 0 < P0 < c 1 + ae−bt

In the following example we determine a logistic growth model for a coyote population. EXAMPLE 7

»

Find and Use a Logistic Model

At the beginning of 2005, the coyote population in a wilderness area was estimated at 200. By the beginning of 2007, the coyote population had increased to 250. A park ranger estimates that the carrying capacity of the wilderness area is 500 coyotes. a.

Use the given data to determine the growth rate constant for the logistic model of this coyote population.

b.

Use the logistic model determined in a. to predict the year in which the coyote population will first reach 400.

Solution a.

If we represent the beginning of the year 2005 by t 苷 0, then the beginning of the year 2007 will be represented by t 苷 2. In the logistic model, make the following substitutions: P共2兲苷 250, c 苷 500, and c P0 500 200 a苷苷苷 1.5. P0 200

Continued 䉴

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Exponential and Logarithmic Functions c 1 aebt 500 P共2兲苷 1 1.5eb2 500 250 苷 1 1.5eb2 250共1 1.5eb2兲苷 500 P共t兲苷

1 1.5eb2 苷

• Substitute the given values. • P (2) 250 • Solve for the growth rate constant b.

500 250

1.5eb2 苷 2 1 1 eb2 苷 1.5 1 2b 苷 ln 1.5 1 1 b 苷 ln 2 1.5 b ⬇ 0.20273255

冉冊冉冊

• Write in logarithmic form.

Using a 苷 1.5, b 苷 0.20273255, and c 苷 500 gives us the following logistic model. 500 P共t兲苷 1 1.5e0.20273255t To determine during what year the logistic model predicts the coyote population will first reach 400, replace P共t兲 with 400 and solve for t.

b.

500 1 1.5e0.20273255t 400共1 1.5e0.20273255t兲苷 500 500 1 1.5e0.20273255t 苷 400 0.20273255t 1.5e 苷 1.25 1 0.25 e0.20273255t 苷 1.5 0.25 0.20273255t 苷 ln 1.5 400 苷

P(t) 500

Coyote population

400

(8.8, 400)

300

冉冊

200 100

10

20

30

t

Year (t = 0 represents the beginning of the year 2005)

P(t) =

冉冊

1 0.25 ln 0.20273255 1.5 ⬇ 8.8

t苷

• Write in logarithmic form. • Solve for t.

According to the logistic model, the coyote population will reach 400 about 8.8 years after the beginning of 2005, which is during the year 2013. The graph of the logistic model is shown in Figure 7.30. Note that P共8.8兲 ⬇ 400 and that as t l , P共t兲 l 500.

500 1 + 1.5e−0.20273255t

Figure 7.30

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Try Exercise 38, page 518

7.5

Exponential Growth and Decay

515

In Example 8, we use a function of the form v 苷 a共1 ekt 兲 to model the velocity of an object that has been dropped from a high elevation. EXAMPLE 8

»

Application to Air Resistance

Assuming that air resistance is proportional to the velocity of a falling object, the velocity (in feet per second) of the object t seconds after it has been dropped is given by v 苷 82共1 e0.39t兲. a.

Determine when the velocity will be 70 feet per second.

b.

The graph of v has v 苷 82 as a horizontal asymptote. Explain the meaning of this asymptote in the context of this example.

ALGEBRAIC SOLUTION

VISUALIZE THE SOLUTION

0.39t

v 苷 82共1 e 兲 0.39t 70 苷 82共1 e 兲 70 苷 1 e0.39t 82 70 e0.39t 苷 1 82 6 0.39t 苷 ln 41 ln共6兾41兲 t苷 ⬇ 4.9277246 0.39

a.

a. • Substitute 70 for v. • Divide each side by 82.

90

• Solve for

e0.39t.

• Write in logarithmic form. • Solve for t.

The velocity will be 70 feet per second after approximately 4.9 seconds. The horizontal asymptote v 苷 82 means that as time increases, the velocity of the object will approach, but never reach or exceed, 82 feet per second.

b.

A graph of y 苷 82共1 e0.39x兲 and y 苷 70 shows that the x-coordinate of the point of intersection is about 4.9.

0 X=4.9 0

Y=70

15

y 苷 82共1 e0.39x兲

Note: The x value shown is rounded to the nearest tenth.

»

Try Exercise 48, page 519

Topics for Discussion 1. Explain the difference between compound interest and simple interest. 2. What is an exponential growth function? Give an example of an application for which an exponential growth function might be an appropriate model. 3. What is an exponential decay function? Give an example of an application for which an exponential decay function might be an appropriate model. 4. Consider the exponential function P共t兲苷 P0e kt and the logistic function c . Explain the similarities and differences between the graphs P共t兲苷 1 aebt of the two functions.

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Exercise Set 7.5 4 micrograms of sodium-24 is injected into a person, the amount A in micrograms remaining in that person after t hours is given by the equation A 苷 4e0.046t.

1. POPULATION GROWTH The number of bacteria N共t兲 present in a culture at time t hours is given by N共t兲苷 2200共2兲t

a. Graph this equation.

Find the number of bacteria present when a. t 苷 0 hours

b. t 苷 3 hours

b. What amount of sodium-24 remains after 5 hours?

2. POPULATION GROWTH The population of a city grows exponentially according to the function

c. What is the half-life of sodium-24? d. In how many hours will the amount of sodium-24 be 1 microgram?

f 共t兲苷 12,400共1.14兲t for 0 t 5 years. Find, to the nearest hundred, the population of the city when t is a. 3 years

b. 4.25 years

3. POPULATION GROWTH A city had a population of 22,600 in 1997 and a population of 24,200 in 2002. a. Find the exponential growth function for the city. Use t 苷 0 to represent the year 1997.

In Exercises 8–12, use the half-life information from Table 7.11, page 508, to work each exercise.

8.

RADIOACTIVE DECAY Find the decay function for the amount of polonium 共210Po兲 that remains in a sample after t days.

9.

GEOLOGY Geologists have determined that Crater Lake in Oregon was formed by a volcanic eruption. Chemical analysis of a wood chip assumed to be from a tree that died during the eruption has shown that it contains approximately 45% of its original carbon-14. Estimate how long ago the volcanic eruption occurred.

10.

RADIOACTIVE DECAY Estimate the percentage of polonium 共210Po兲 that remains in a sample after 2 years. Round to the nearest hundredth of a percent.

11.

ARCHEOLOGY The Rhind papyrus, named after A. Henry Rhind, contains most of what we know today of ancient Egyptian mathematics. A chemical analysis of a sample from the papyrus has shown that it contains approximately 75% of its original carbon-14. Estimate the age of the Rhind papyrus.

b. Use the growth function to predict the population of the city in 2012. Round to the nearest hundred. 4. POPULATION GROWTH A city had a population of 53,700 in 1999 and a population of 58,100 in 2003. a. Find the exponential growth function for the city. Use t 苷 0 to represent the year 1999. b. Use the growth function to predict the population of the city in 2011. Round to the nearest hundred. 5.

6.

POPULATION GROWTH The population of Miami, Florida, is growing exponentially. The population of Miami was 362,300 in the year 2000 and 379,700 in 2004. Find the exponential growth function that models the population growth of Miami and use it to predict the population of Miami in 2009. Use t 苷 0 to represent the year 2000. Round to the nearest hundred. POPULATION GROWTH The population of Aurora, Colorado, is growing exponentially. The population of Aurora was 276,400 in the year 2000 and 291,800 in 2004. Find the exponential growth function that models the population growth of the Aurora and use it to predict the population of Aurora in 2008. Use t 苷 0 to represent the year 2000. Round to the nearest hundred.

7. MEDICINE Sodium-24 is a radioactive isotope of sodium that is used to study circulatory dysfunction. Assuming that

12. ARCHEOLOGY Estimate the age of a bone if it now contains 65% of its original amount of carbon-14. Round to the nearest 100 years. 13. COMPOUND INTEREST If $8000 is invested at an annual interest rate of 5% and compounded annually, find the balance after a. 4 years

b. 7 years

14. COMPOUND INTEREST If $22,000 is invested at an annual interest rate of 4.5% and compounded annually, find the balance after a. 2 years

b. 10 years

7.5 15. COMPOUND INTEREST If $38,000 is invested at an annual interest rate of 6.5% for 4 years, find the balance if the interest is compounded a. annually

b. daily

c. hourly

16. COMPOUND INTEREST If $12,500 is invested at an annual interest rate of 8% for 10 years, find the balance if the interest is compounded a. annually

b. daily

c. hourly

Exponential Growth and Decay

517

27. P共t兲苷

157,500 1 2.5e0.04t

28. P共t兲苷

51 1 1.04e0.03t

29. P共t兲苷

2400 1 7e0.12t

30. P共t兲苷

320 1 15e0.12t

In Exercises 31 to 34, use algebraic procedures to find the logistic growth model for the data.

31. P0 苷 400, P共2兲苷 780, and the carrying capacity is 5500.

17. COMPOUND INTEREST Find the balance if $15,000 is invested at an annual rate of 10% for 5 years, compounded continuously. 18. COMPOUND INTEREST Find the balance if $32,000 is invested at an annual rate of 8% for 3 years, compounded continuously. 19. COMPOUND INTEREST How long will it take $4000 to double if it is invested in a certificate of deposit that pays 7.84% annual interest compounded continuously? Round to the nearest tenth of a year. 20. COMPOUND INTEREST How long will it take $25,000 to double if it is invested in a savings account that pays 5.88% annual interest compounded continuously? Round to the nearest tenth of a year. 21. CONTINUOUS COMPOUNDING INTEREST Use the Continuous Compounding Interest Formula to derive an expression for the time it will take money to triple when invested at an annual interest rate of r compounded continuously. 22. CONTINUOUS COMPOUNDING INTEREST How long will it take $1000 to triple if it is invested at an annual interest rate of 5.5% compounded continuously? Round to the nearest year. 23. CONTINUOUS COMPOUNDING INTEREST How long will it take $6000 to triple if it is invested in a savings account that pays 7.6% annual interest compounded continuously? Round to the nearest year.

32. P0 苷 6200, P共8兲苷 7100, and the carrying capacity is 9500. 33. P0 苷 18, P共3兲苷 30, and the carrying capacity is 100. 34. P0 苷 3200, P共22兲 ⬇ 5565, and the growth rate constant is 0.056. 35. REVENUE The annual revenue R, in dollars, of a new company can be closely modeled by the logistic function R共t兲苷

625,000 1 3.1e0.045t

where the natural number t is the time, in years, since the company was founded. a. According to the model, what will be the company’s annual revenue for its first year and its second year (t 苷 1 and t 苷 2) of operation? Round to the nearest $1000. b. According to the model, what will the company’s annual revenue approach in the long-term future? 36. NEW CAR SALES The number of cars A sold annually by an automobile dealership can be closely modeled by the logistic function A共t兲苷

1650 1 2.4e0.055t

where the natural number t is the time, in years, since the dealership was founded.

24. CONTINUOUS COMPOUNDING INTEREST How long will it take $10,000 to triple if it is invested in a savings account that pays 5.5% annual interest compounded continuously? Round to the nearest year.

a. According to the model, what number of cars will the dealership sell during its first year and its second year (t 苷 1 and t 苷 2) of operation? Round to the nearest unit.

In Exercises 25 to 30, determine the following constants for the given logistic growth model. a. The carrying capacity b. The growth rate constant c. The initial population P0

b. According to the model, what will the dealership’s annual car sales approach in the long-term future?

25. P共t兲苷

1900 1 8.5e0.16t

26. P共t兲苷

32,550 1 0.75e0.08t

37. POPULATION GROWTH The population of wolves in a preserve satisfies a logistic model in which P0 苷 312 in the year 2005, c 苷 1600, and P共6兲苷 416.

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a. Determine the logistic model for this population, where t is the number of years after 2005. b. Use the logistic model from a. to predict the size of the wolf population in 2015. 38. POPULATION GROWTH The population of groundhogs on a ranch satisfies a logistic model in which P0 苷 240 in the year 2004, c 苷 3400, and P共1兲苷 310. a. Determine the logistic model for this population, where t is the number of years after 2004. b. Use the logistic model from a. to predict the size of the groundhog population in 2011. 39. POPULATION GROWTH The population of squirrels in a nature preserve satisfies a logistic model in which P0 苷 1500 in the year 2007. The carrying capacity of the preserve is estimated at 8500 squirrels, and P共2兲苷 1900. a. Determine the logistic model for this population, where t is the number of years after 2007. b. Use the logistic model from a. to predict the year in which the squirrel population will first exceed 4000. 40. POPULATION GROWTH The population of walruses on an island satisfies a logistic model in which P0 苷 800 in the year 2003. The carrying capacity of the island is estimated at 5500 walruses, and P共1兲苷 900. a. Determine the logistic model for this population, where t is the number of years after 2003. b. Use the logistic model from a. to predict the year in which the walrus population will first exceed 2000.

42. PSYCHOLOGY According to a software company, the users of its typing tutorial can expect to type N共t兲 words per minute after t hours of practice with the product, according to the function N共t兲苷 100共1.04 0.99t兲. a. How many words per minute can a student expect to type after 2 hours of practice? b. How many words per minute can a student expect to type after 40 hours of practice? c. According to the function N, how many hours (to the nearest hour) of practice will be required before a student can expect to type 60 words per minute? 43. PSYCHOLOGY In the city of Whispering Palms, which has a population of 80,000 people, the number of people P共t兲 exposed to a rumor in t hours is given by the function P共t兲苷 80,000共1 e0.0005t兲. a. Find the number of hours until 10% of the population have heard the rumor. b. Find the number of hours until 50% of the population have heard the rumor. 44. LAW A lawyer has determined that the number of people P共t兲 in a city of 1,200,000 people who have been exposed to a news item after t days is given by the function P共t兲苷 1,200,000共1 e0.03t兲 a. How many days after a major crime has been reported have 40% of the population heard of the crime? b. A defense lawyer knows it will be very difficult to pick an unbiased jury after 80% of the population have heard of the crime. After how many days will 80% of the population have heard of the crime?

41. PHYSICS Newton’s Law of Cooling states that if an object at temperature T0 is placed into an environment at constant temperature A, then the temperature of the object, T共t兲 (in degrees Fahrenheit), after t minutes is given by T共t兲苷 A 共T0 A兲ekt, where k is a constant that depends on the object.

45. DEPRECIATION An automobile depreciates according to the function V共t兲苷 V0 共1 r兲t, where V共t兲 is the value in dollars after t years, V0 is the original value, and r is the yearly depreciation rate. A car has a yearly depreciation rate of 20%. Determine, to the nearest 0.1 year, in how many years the car will depreciate to half its original value.

a. Determine the constant k (to the nearest thousandth) for a canned soda drink that takes 5 minutes to cool from 75°F to 65°F after being placed in a refrigerator that maintains a constant temperature of 34°F.

46. PHYSICS The current I共t兲 (measured in amperes) of a circuit is given by the function I共t兲苷 6共1 e2.5t兲, where t is the number of seconds after the switch is closed.

b. What will be the temperature (to the nearest degree) of the soda drink after 30 minutes? c. When (to the nearest minute) will the temperature of the soda drink be 36°F?

a. Find the current when t 苷 0. b. Find the current when t 苷 0.5.

7.5

Exponential Growth and Decay

519

b. Determine, to the nearest 0.1 second, the time it takes the object to fall 50 feet.

c. Solve the equation for t.

Current, in amperes

I(t)

c. Calculate the slope of the secant line through 共1, s共1兲兲 and 共2, s共2兲兲.

6

d. 2

1

3

t

Time, in seconds

47. AIR RESISTANCE Assuming that air resistance is proportional to velocity, the velocity v, in feet per second, of a falling object after t seconds is given by v 苷 32共1 et兲.

The distance s (in feet) that the object in Exercise 48 will fall in t seconds is given by the function s 苷 64t 128共et/2 1兲. a. Graph this equation for t 0.

a. Graph this equation for t 0.

b. Determine, to the nearest 0.1 second, the time it takes the object to fall 50 feet.

b. Determine algebraically, to the nearest 0.01 second, when the velocity is 20 feet per second.

c. Calculate the slope of the secant line through 共1, s共1兲兲 and 共2, s共2兲兲.

c. Determine the horizontal asymptote of the graph of v.

d.

d.

Write a sentence that explains the meaning of the horizontal asymptote in the context of this application.

48. AIR RESISTANCE Assuming that air resistance is proportional to velocity, the velocity v, in feet per second, of a falling object after t seconds is given by v 苷 64共1 et/2兲. a. Graph this equation for t 0. b. Determine algebraically, to the nearest 0.1 second, when the velocity is 50 feet per second. c. Determine the horizontal asymptote of the graph of v. d.

49.

50.

Write a sentence that explains the meaning of the slope of the secant line you calculated in c.

Write a sentence that explains the meaning of the horizontal asymptote in the context of this application.

The distance s (in feet) that the object in Exercise 47 will fall in t seconds is given by the function s 苷 32t 32共et 1兲. a. Graph this equation for t 0.

Write a sentence that explains the meaning of the slope of the secant line you calculated in c.

51. LEARNING THEORY The logistic model is also used in learning theory. Suppose that historical records from employee training at a company show that the percent score on a product information test is given by P苷

100 1 25e0.095t

where t is the number of hours of training. What is the number of hours (to the nearest hour) of training needed before a new employee will answer 75% of the questions correctly? 52. LEARNING THEORY A company provides training in the assembly of a computer circuit to new employees. Past experience has shown that the number of correctly assembled circuits per week can be modeled by N苷

250 1 249e0.503t

where t is the number of weeks of training. What is the number of weeks (to the nearest week) of training needed before a new employee will correctly make 140 circuits?

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Connecting Concepts 53. MEDICATION LEVEL A patient is given three doses of aspirin. Each dose contains 1 gram of aspirin. The second and third doses are each taken 3 hours after the previous dose is administered. The half-life of the aspirin is 2 hours. The amount of aspirin A in the patient’s body t hours after the first dose is administered is

再

0.5t/2 0t3 3t6 A共t兲苷 0.5t/2 0.5共t3兲/2 0.5t/2 0.5共t3兲/2 0.5共t6兲/2 t6 Find, to the nearest hundredth of a gram, the amount of aspirin in the patient’s body when a. t 苷 1

b. t 苷 4

c. t 苷 9

54.

MEDICATION LEVEL Use the dosage formula in Exercise 53 to determine when, to the nearest tenth of an hour, the amount of aspirin in the patient’s body first reaches 0.25 gram.

55.

ANNUAL GROWTH RATE The exponential growth function for the population of a city is N共t兲苷 78,245e0.0245t, where t is in years. Because

we can write the growth function as

冉

N共t兲苷 78,245共1.0248兲t ⬇ 78,245 1

冊

0.0248 1

1t

In this form we can see that the city’s population is growing by 2.48% per year. The population of the city of Lake Tahoe, Nevada, can be modeled by the exponential growth function N共t兲苷 22,755e0.0287t. Find the annual growth rate, expressed as a percent, of Lake Tahoe. Round to the nearest hundredth of a percent. 56. OIL SPILLS Crude oil leaks from a tank at a rate that depends on the amount of oil that remains in the tank. 1 Because of the oil in the tank leaks out every 2 hours, 8 the volume V共t兲 of oil in the tank after t hours is given by V共t兲苷 V0 共0.875兲t/2, where V0 苷 350,000 gallons is the number of gallons in the tank at the time the tank started to leak 共t 苷 0兲. a. How many gallons does the tank hold after 3 hours? b. How many gallons does the tank hold after 5 hours?

e0.0245t 苷共e0.0245 兲t ⬇ 共1.0248兲t

c. How long, to the nearest hour, will it take until 90% of the oil has leaked from the tank?

Projects A DECLINING LOGISTIC MODEL If P0 c (which implies that c de1 a 0), then the logistic function P共t兲苷 1 aebt creases as t increases. Biologists often use this type of logistic function to model populations that decrease over time. See the following figure.

P共t兲苷

1000 1 共0.3333兲e0.05t

where t is the time, in years, since the lake was first stocked with fish. a. What was the fish population when the lake was first stocked with fish?

P(t)

b. According to the logistic model, what will the fish population approach in the long-term future? P0

2. A DECLINING DEER POPULATION The deer population in a reserve is given by the logistic function

c

P共t兲苷 t

P(t) =

c , P0 > c 1 + ae−bt

1. A DECLINING FISH POPULATION A biologist finds that the fish population in a small lake can be closely modeled by the logistic function

1800 1 共0.25兲e0.07t

where t is the time, in years, since July 1, 2001. a. What was the deer population on July 1, 2001? What was the deer population on July 1, 2003? b. According to the logistic model, what will the deer population approach in the long-term future?

7.6 3.

MODELING WORLD RECORD TIMES IN THE MEN’S MILE RACE In the early 1950s, many people speculated that no runner would ever run a mile race in under 4 minutes. During the period from 1913 to 1945, the world record in the mile event had been reduced from 4.14.4 (4 minutes, 14.4 seconds) to 4.01.4, but no one seemed capable of running a sub-four-minute mile. Then, in 1954, Roger Bannister broke through the four-minute barrier by

Section 7.6 ■ ■ ■

Modeling Data with Exponential and Logarithmic Functions

Analyze Scatter Plots Model Data Find a Logistic Growth Model

521

running a mile in 3.59.6. In 1999, the current record of 3.43.13 was established. It is fun to think about future record times in the mile race. Will they ever go below 3 minutes, 30 seconds? Below 3 minutes, 20 seconds? What about a sub-three-minute mile? A declining logistic function that closely models the world record times WR, in seconds, in the men’s mile run from 1913 共t 苷 0兲 to 1999 共t 苷 86兲 is given by WR共t兲苷

199.13 1 共0.21726兲e0.0079889t

a. Use the above logistic function to predict the world record time for the men’s mile run in the year 2020 and the year 2050. b. According to the logistic function, what time will the world record in the men’s mile event approach but never break through?

Modeling Data with Exponential and Logarithmic Functions P R E PA R E F O R T H I S S E C T I O N Prepare for this section by completing the following exercises. The answers can be found on page A40.

PS1. Determine whether N共t兲苷 4 ln t is an increasing or a decreasing function. [7.2] PS2. Determine whether P共t兲苷 1 2共1.05t兲 is an increasing or a decreasing function. [7.1] PS3. Evaluate P共t兲苷

108 for t 苷 0. [7.1] 1 2e0.1t

PS4. Evaluate N共t兲苷 840e1.05t for t 苷 0. [7.1] PS5. Solve 10 苷

20 for t. Round to the nearest tenth. [7.4] 1 2.2e0.05t

PS6. Determine the horizontal asymptote of the graph of P共t兲苷

55 , for t 0. [7.5] 1 3e0.08t

Analyze Scatter Plots In Section 1.7 we used linear and quadratic functions to model several data sets. However, in some applications, data can be modeled more closely by using

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Chapter 7

Exponential and Logarithmic Functions exponential or logarithmic functions. For instance, Figure 7.31 illustrates some scatter plots that can be modeled effectively by exponential and logarithmic functions. y

y

y

y

x

x

a. Exponential increasing: y = abx, a > 0, b > 1

x

b. Exponential decreasing y = abx, a > 0, 0 < b < 1

x

c. Logarithmic increasing: y = a + b ln x, b > 0

d. Logarithmic decreasing: y = a + b ln x, b < 0

Figure 7.31 Exponential and Logarithmic Models

The terms concave upward and concave downward are often used to describe a graph. For instance, Figures 7.32a and 7.32b show the graphs of two increasing functions that join the points P and Q. The graphs of f and g differ in that they bend in different directions. We can distinguish between these two types of “bending” by examining the positions of tangent lines to the graphs. In Figures 7.32c and 7.32d, tangent lines (in red) have been drawn to the graphs of f and g. The graph of f lies above its tangent lines and the graph of g lies below its tangent lines. The function f is said to be concave upward, and g is concave downward. a. y

b. y Q

Q

f g P

P x2 x

x1

c. f is concave upward.

x2 x

x1

d. g is concave downward. y

y

Q

Q f g P

P

x1

x2 x

x1

x2 x

Figure 7.32

Definition of Concavity If the graph of f lies above all of its tangents on an interval 关x1, x2兴, then f is concave upward on 关x1, x2兴. If the graph of f lies below all of its tangents on an interval 关x1, x2兴, then f is concave downward on 关x1, x2兴.

7.6

523

Modeling Data with Exponential and Logarithmic Functions

An examination of the graphs in Figure 7.31 shows that the graphs of all exponential functions of the form y 苷 abx, a 0, b 0, b 苷 1 are concave upward. The graphs of increasing logarithmic functions of the form y 苷 a b ln x, b 0 are concave downward, and the graphs of decreasing logarithmic functions of the form y 苷 a b ln x, b 0, are concave upward. In Example 1 we analyze scatter plots to determine whether the shape of the scatter plot can be best approximated by a function that is concave upward or concave downward.

QUESTION

Is the graph of y 苷 5 2 ln x concave upward or concave downward?

EXAMPLE 1

»

Analyze Scatter Plots

For each of the following data sets, determine whether the most suitable model of the data would be an increasing exponential function or an increasing logarithmic function. TO REVIEW

See section 1.7 if you need to review the steps needed to create a scatter plot on a TI-83/TI-83 Plus/TI-84 Plus calculator.

A 苷兵共1, 0.6兲, 共2, 0.7兲, 共2.8, 0.8兲, 共4, 1.3兲, 共6, 1.5兲, 共6.5, 1.6兲, 共8, 2.1兲, 共11.2, 4.1兲, 共12, 4.6兲, 共15, 8.2兲其 B 苷兵共1.5, 2.8兲, 共2, 3.5兲, 共4.1, 5.1兲, 共5, 5.5兲, 共5.5, 5.7兲, 共7, 6.1兲, 共7.2, 6.4兲, 共8, 6.6兲, 共9, 6.9兲, 共11.6, 7.4兲, 共12.3, 7.5兲, 共14.7, 7.9兲其

Solution For each set, construct a scatter plot of the data. See Figure 7.33. 10

10

20

0 0

16

0 0

Scatter plot of A

Scatter plot of B

Figure 7.33

The scatter plot of A suggests that A is an increasing function that is concave upward. Thus an increasing exponential function would be the most suitable model for data set A. The scatter plot of B suggests that B is an increasing function that is concave downward. Thus an increasing logarithmic function would be the most suitable model for data set B.

»

Try Exercise 4, page 529

ANSWER

The graph of y 苷 5 2 ln x is concave upward because the b-value, 2, is less than zero. See Figure 7.31d.

524

Chapter 7

Exponential and Logarithmic Functions

Model Data

Integrating Technology Most graphing utilities have built-in routines that can be used to determine the exponential or logarithmic regression function that models a set of data. On a TI-83/TI-83 Plus/TI-84 Plus, the ExpReg instruction is used to find the exponential regression function and the LnReg instruction is used to find the logarithmic regression function. The TI-83/TI-83 Plus/TI-84 Plus does not show the value of the regression coefficient r or the coefficient of determination unless the DiagnosticOn command has been entered. The DiagnosticOn command is in the CATALOG menu.

take note The value of a diamond is generally

The methods used to model data using exponential or logarithmic functions are similar to the methods used in Section 1.7 to model data using linear or quadratic functions. Here is a summary of the modeling process. The Modeling Process Use a graphing utility to: 1. Construct a scatter plot of the data to determine which type of function will effectively model the data. 2. Find the equation of the modeling function and the correlation coefficient or the coefficient of determination for the equation. 3. Examine the correlation coefficient or the coefficient of determination and view a graph that displays both the modeling function and the scatter plot to determine how well your function fits the data.

In the following example we use the modeling process to find an exponential function that closely models the value of a diamond as a function of its weight. EXAMPLE 2

»

Model an Application with an Exponential Function

A diamond merchant has determined the values of several white diamonds that have different weights (measured in carats), but are similar in quality. See Table 7.13. Table 7.13 0.50 ct

0.75 ct

1.00 ct

1.25 ct

1.50 ct

1.75 ct

2.00 ct

3.00 ct

4.00 ct

$4,600

$5,000

$5,800

$6,200

$6,700

$7,300

$7,900

$10,700

$14,500

Find a function that models the values of the diamonds as a function of their weights, and use the function to predict the value of a 3.5-carat diamond of similar quality.

determined by its color, cut, clarity, and carat weight. These

Solution

characteristics of a diamond are

1. Construct a scatter plot of the data.

known as the four c’s. In Example 2

18,000

enables us to model the value of

L1 .5 .75 1 1.25 1.5 1.75 2

each diamond as a function of just

L2(6) =7300

we have assumed that the color, cut, and clarity of all the diamonds are similar. This assumption

its carat weight.

L2 4600 5000 5800 6200 6700 7300 7900

L3

2

0

5 0

Figure 7.34

7.6

Modeling Data with Exponential and Logarithmic Functions

From the scatter plot in Figure 7.34, it appears that the data can be closely modeled by an exponential function of the form y 苷 abx, a 0 and b 1.

Math Matters The Hope Diamond, shown below, is the world’s largest deep blue diamond. It has a weight of 45.52 carats. We should not expect the function y ⬇ 4067.6 1.3816x in Example 2 to yield an accurate value of the Hope Diamond because the Hope Diamond is not the same type of diamond as the diamonds in Table 7.13, and its weight is much larger than the weights of the diamonds in Table 7.13.

525

2. Find the equation of the model. The calculator display in Figure 7.35 shows that the exponential regression equation is y ⬇ 4067.6共1.3816兲x, where x is the carat weight of the diamond and y is the value of the diamond.

ExpReg y=a*b^x a=4067.641145 b=1.381644186 r2=.994881215 r=.9974373238

Figure 7.35 ExpReg display (DiagnosticOn)

3. Examine the correlation coefficient or the coefficient of determination. The correlation coefficient r ⬇ 0.9974 is close to 1. This indicates that the exponential regression function y ⬇ 4067.6共1.3816兲x provides a good fit for the data. The graph in Figure 7.36 also shows that the exponential regression function provides a good model for the data. 18,000

5

0

The Hope Diamond is on display at the Smithsonian Museum of Natural History in Washington, D.C.

0

Figure 7.36

To estimate the value of a 3.5-carat diamond, substitute 3.5 for x in the exponential regression function or use the VALUE command in the CALCULATE menu to evaluate the exponential regression function at x 苷 3.5. See Figure 7.37.

18,000

Y1=4067.641144727* 1.381…

0 X=3.5 0

Y=12610.415

5

Figure 7.37

According to the exponential regression function, the value of a 3.5-carat diamond of similar quality is about $12,610.

»

Try Exercise 22, page 531

When you are selecting a function to model a given set of data, try to find a function that provides a good fit to the data and is likely to produce realistic predictions. The following guidelines may facilitate the selection process.

526

Chapter 7

Exponential and Logarithmic Functions Guidelines for Selecting a Modeling Function 1. Use a graphing utility to construct a scatter plot of the data. 2. Compare the graphical features of the scatter plot with the graphical features of the basic modeling functions available on the graphing utility: linear, quadratic, cubic, exponential, logarithmic, or logistic. Pay particular attention to the concave nature of each function. Eliminate those functions that do not display the desired concavity. 3. Use the graphing utility to find the equation of each type of function you identified in Step 2 as a possible model. 4. Determine how well each function fits the given data, and compare the graphs of the functions to determine which function is most likely to produce realistic predictions.

EXAMPLE 3

»

Select a Modeling Function and Make a Prediction

Table 7.14 shows the winning times in the women’s Olympic

100-meter freestyle event for the years 1960 to 2004. Table 7.14 Women’s Olympic 100-Meter Freestyle, 1960 to 2004 Year

Time (in seconds)

Year

Time (in seconds)

1960

61.2

1984

55.92

1964

59.5

1988

54.93

1968

60.0

1992

54.64

1972

58.59

1996

54.50

1976

55.65

2000

53.83

1980

54.79

2004

53.84

Source: Time Almanac 2006.

take note When you use a graphing utility to find a logarithmic model, remember that the domain of y 苷 a b ln x is the set of positive numbers. Thus zero must not be used as an x value of a data point. This is the reason we have used x 1 to represent the year 1960 in Example 3.

Find a function to model the data, and use the function to predict the winning time in the women’s Olympic 100-meter freestyle event for the year 2012.

Solution Construct a scatter plot of the data. See Figure 7.38. (Note: This scatter plot was produced using x 苷 1 to represent the year 1960, x 苷 2 to represent the year 1964, . . ., and x 苷 12 to represent the year 2004.) The general shape of the scatter plot suggests that we consider functions whose graphs are decreasing and concave upward. Thus we

65

0

20 45

Figure 7.38

7.6

527

Modeling Data with Exponential and Logarithmic Functions consider a decreasing exponential function and a decreasing logarithmic function as possible models. Use a graphing utility to find the exponential regression function and the logarithmic regression function for the data. See Figures 7.39 and 7.40.

ExpReg y=a*b^x a=60.85756911 b=.988350233 r2=.849501015 r=-.921683794

LnReg y=a+blnx a=61.93339969 b= -3.292644718 r2=.898941686 r= -.948125353

Figure 7.39

Figure 7.40

The exponential function is y ⬇ 60.85757(0.98835)x and the logarithmic function is y ⬇ 61.93340 3.29264 ln x. The coefficient of determination r 2 for the logarithmic regression is larger than the coefficient of determination for the exponential regression. See Figures 7.39 and 7.40. Thus the logarithmic regression function provides a better fit to the data than the exponential regression function. The correlation coefficients r can also be used to determine which function provides the better fit. For decreasing functions, the function with correlation coefficient closest to 1 provides the better fit.

Notice that the graph of the logarithmic function has the desired behavior to the right of the scatter plot. That is, it is a gradually decreasing curve, and this is the general behavior we would expect for future winning times in the 100-meter freestyle event. The graph of the exponential function is almost linear and is decreasing at a rapid pace, which is not what we would expect for results in an established Olympic event. See Figure 7.41. Thus we select the logarithmic function as our modeling function.

To predict the winning time for this event in the year 2012 (represented by x 苷 14), substitute 14 for x in the equation of the logarithmic function or use the VALUE command in the CALCULATE menu to produce the approximate time of 53.24 seconds, as shown in Figure 7.42.

65

LnReg

ExpReg 0

20 45

Figure 7.41

65

Y2=61.933399686+ 3 2926

0 X=14 45

Y Y=53.243922 20

Figure 7.42

»

Try Exercise 24, page 531

528

Chapter 7

Exponential and Logarithmic Functions

Find a Logistic Growth Model If a scatter plot of a set of data suggests that the data can be effectively modeled by a logistic growth model, then you can use the Logistic feature of a graphing utility to find the logistic growth model. This process is illustrated in Example 4. EXAMPLE 4

»

Find a Logistic Growth Model

Table 7.15 shows the population of deer in an animal preserve for the

years 1990 to 2004. Table 7.15 Deer Population at the Wild West Animal Preserve Year

Population

Year

Population

Year

Population

1990

320

1995

1150

2000

2620

1991

410

1996

1410

2001

2940

1992

560

1997

1760

2002

3100

1993

730

1998

2040

2003

3300

1994

940

1999

2310

2004

3460

Find a logistic model that approximates the deer population as a function of the year. Use the model to predict the deer population in the year 2010.

Solution

Integrating Technology On a TI-83/TI-83 Plus/ TI-84 Plus graphing calculator, the logistic growth model is given in the form y苷

c 1 aebx

Think of the variable x as the time t and the variable y as P共t兲.

1. Construct a scatter plot of the data. Enter the data into a graphing utility, and then use the utility to display a scatter plot of the data. In this example we represent the year 1990 by x 苷 0, the year 2004 by x 苷 14, and the deer population by y. Figure 7.43 shows that the data can be closely approximated by a logistic growth model. 2. Find the equation of the model. On a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, select B: Logistic, which is in the STAT CALC menu. The logistic function for the data is 3965.3 y⬇ . See Figure 7.44. 1 11.445e0.31152x

4800

0

24 0

Figure 7.43

Logistic y=c/(1+ae^(-bx)) a=11.44466821 b=.3115234553 c=3965.337214

Figure 7.44

3. Examine the fit. A TI-83/TI-83 Plus/TI-84 Plus calculator does not compute the coefficient of determination or the correlation coefficient for a logistic model. However, Figure 7.45 shows that the logistic model provides a good fit to the data. The VALUE command in the CALCULATE menu shows that the logistic model predicts a deer population of about 3878 in the year 2010 (x 苷 20). See Figure 7.46.

7.6

Modeling Data with Exponential and Logarithmic Functions

529

4800

4800

Y1=3965.3372136266/(1+1…

0

0 X=20 0

24 0

Figure 7.45

Y Y=3877.9698 24

Figure 7.46

»

Try Exercise 26, page 532

Topics for Discussion 1. A student tries to determine the exponential regression equation for the following data. x

1

2

3

4

5

y

8

2

0

1.5

2

The student’s calculator displays an ERROR message. Explain why the calculator was unable to determine the exponential regression equation for the data. 2. Consider the logarithmic model h共x兲苷 6 2 ln x. a.

Is h an increasing or a decreasing function?

b. Is h concave up or concave down on the interval 共0, 兲? c.

Find, if possible, h共0兲 and h共e兲.

d. Does h have a horizontal asymptote? Explain.

Exercises Set 7.6 In Exercises 1 to 6, use a scatter plot of the given data to determine which of the following types of functions might provide a suitable model of the data. ■

An increasing exponential function y 苷 abx, a 0, b 1 (See Figure 7.31a.)

(Note: Some data sets can be closely modeled by more than one type of function.)

1. 兵共1, 3兲, 共1.5, 4兲, 共2, 6兲, 共3, 13兲, 共3.5, 19兲, 共4, 27兲其 2. 兵共1.0, 1.12兲, 共2.1, 0.87兲, 共3.2, 0.68兲, 共3.5, 0.63兲, 共4.4, 0.52兲其

■

An increasing logarithmic function y 苷 a b ln x, b 0 (See Figure 7.31c.)

3. 兵共1, 2.4兲, 共2, 1.1兲, 共3, 0.5兲, 共4, 0.2兲, 共5, 0.1兲其

■

A decreasing exponential function y 苷 abx, a 0, 0 b 1 (See Figure 7.31b.)

4. 兵共5, 2.3兲, 共7, 3.9兲, 共9, 4.5兲, 共12, 5.0兲, 共16, 5.4兲, 共21, 5.8兲, 共26, 6.1兲其

■

A decreasing logarithmic function y 苷 a b ln x, b 0 (See Figure 7.31d.)

5. 兵共1, 2.5兲, 共1.5, 1.7兲, 共2, 0.7兲, 共3, 0.5兲, 共3.5, 1.3兲, 共4, 1.5兲其 6. 兵共1, 3兲, 共1.5, 3.8兲, 共2, 4.4兲, 共3, 5.2兲, 共4, 5.8兲, 共6, 6.6兲其

530

Chapter 7

Exponential and Logarithmic Functions the year 1994 and t 苷 2 to represent the year 1996. Round the constants a and b to the nearest hundred thousandth. State the correlation coefficient r for each model.

In Exercises 7 to 10, find the exponential regression function for the data. State the correlation coefficient r. Round a, b, and r to the nearest hundred thousandth.

7. 兵共10, 6.8兲, 共12, 6.9兲, 共14, 15.0兲, 共16, 16.1兲, 共18, 50.0兲, 共19, 20.0兲其

b. Examine the correlation coefficients to determine which model provides a better fit for the data.

8. 兵共2.6, 16.2兲, 共3.8, 48.8兲, 共5.1, 160.1兲, 共6.5, 590.2兲, 共7, 911.2兲其 9. 兵共0, 1.83兲, 共1, 0.92兲, 共2, 0.51兲, 共3, 0.25兲, 共4, 0.13兲, 共5, 0.07兲其

c. Use the model you selected in b. to predict the average U.S. movie theater ticket price for the year 2010.

10. 兵共4.5, 1.92兲, 共6.0, 1.48兲, 共7.5, 1.14兲, 共10.2, 0.71兲, 共12.3, 0.49兲其 In Exercises 11 to 14, find the logarithmic regression function for the data. State the correlation coefficient r. Round a, b, and r to the nearest hundred thousandth.

20.

11. 兵共5, 2.7兲, 共6, 2.5兲, 共7.2, 2.2兲, 共9.3, 1.9兲, 共11.4, 1.6兲, 共14.2, 1.3兲其

GENERATION OF GARBAGE According to the U.S. Environmental Protection Agency, the amount of garbage generated per person has been increasing over the last few decades. The following table shows the per capita garbage, in pounds per day, generated in the United States.

12. 兵共11, 15.75兲, 共14, 15.52兲, 共17, 15.34兲, 共20, 15.18兲, 共23, 15.05兲其 13. 兵共3, 16.0兲, 共4, 16.5兲, 共5, 16.9兲, 共7, 17.5兲, 共8, 17.7兲, 共9.8, 18.1兲其 14. 兵共8, 67.1兲, 共10, 67.8兲, 共12, 68.4兲, 共14, 69.0兲, 共16, 69.4兲其

Year Pounds per day, p

In Exercises 15 to 18, find the logistic regression function for the data. Round the constants a, b, and c to the nearest hundred thousandth.

1960

1970

1980

1990

2003

2.7

3.3

3.7

4.5

4.5

a. Find a linear model and a logarithmic model for the data. Use t as the independent variable (domain) and p as the dependent variable (range). Represent the year 1960 by t 苷 60 and the year 1970 by t 苷 70.

15. 兵共0, 81兲, 共2, 87兲, 共6, 98兲, 共10, 110兲, 共15, 125兲其 16. 兵共0, 175兲, 共5, 195兲, 共10, 217兲, 共20, 264兲, 共35, 341兲其

b. Examine the correlation coefficients to determine which model provides a better fit for the data.

17. 兵共0, 955兲, 共10, 1266兲, 共20, 1543兲, 共30, 1752兲其 c. Use the model you selected in b. to predict the amount of garbage that will be generated per capita per day in 2009. Round to the nearest tenth of a pound.

18. 兵共0, 1588兲, 共5, 2598兲, 共10, 3638兲, 共25, 5172兲其 19.

MOVIE TICKET PRICES The following table shows the average U.S. movie theater ticket prices for selected years from 1994 to 2004.

21.

HYPOTHERMIA The following table shows the time T, in hours, before a scuba diver wearing a 3-millimeter-thick wet suit reaches hypothermia (95 F) for various water temperatures F, in degrees Fahrenheit.

Year

Price, P

1994

$4.08

1996

$4.42

1998

$4.69

Water temperature, F

Time T, hours

2000

$5.39

41

1.1

2002

$5.80

46

1.4

2004

$6.21

50

1.8

59

3.7

Source: National Association of Theatre Owners

a. Determine an exponential regression model and a linear regression model for the data. Use t 苷 0 to represent

a. Find an exponential regression model for the data. Round the constants a and b to the nearest hundred thousandth.

7.6

b. Use the model from a. to estimate the time it takes for the diver to reach hypothermia in water that has a temperature of 65 F. Round to the nearest tenth of an hour.

23.

an hour. How much greater is this result compared with the answer to Exercise 21b.? 24.

ATMOSPHERIC PRESSURE The following table shows the earth’s atmospheric pressure y (in newtons per square centimeter) at an altitude of x kilometers. Find a suitable function that models the atmospheric pressure as a function of the altitude. Use the function to estimate the atmospheric pressure at an altitude of 24 kilometers. Round to the nearest tenth of a newton per square centimeter.

400-METER RACE The following table lists the progression of world record times in the men’s 400-meter race for the years from 1948 to 2005. (Note: No new world record times were set during the time period from 2000 to 2005.) World Record Times in the Men’s 400-Meter Race, 1948 to 2005

Year

Time, in seconds

Year

Time, in seconds

1948

45.9

1964

44.9

1950

45.8

1967

44.5

Altitude x, kilometers

Pressure y, newtons/cm2

0

10.3

1955

45.4

1968

44.1

2

8.0

1956

45.2

1968

43.86

4

6.4

1960

44.9

1988

43.29

6

5.1

1963

44.9

1999

43.18

8

4.0

10

3.2

12

2.5

14

2.0

16

1.6

18

1.3

HYPOTHERMIA The following table shows the time T, in hours, before a scuba diver wearing a 4-millimeter-thick wet suit reaches hypothermia (95 E) for various water temperatures F, in degress Fahrenheit. Water temperature, F

Time T, hours

41

1.5

46

1.9

50

2.4

59

5.2

a. Find an exponential regression model for the data. Round the constants a and b to the nearest hundred thousandth. b. Use the model from a. to estimate the time it takes for the diver to reach hypothermia in water that has a temperature of 65 F. Round to the nearest tenth of

Source: Track and Field Statistics, http://trackfield.brinkster.net/Main.asp.

a. Determine whether the data can best be modeled by a decreasing exponential function or a decreasing logarithmic function. Let x 苷 48 represent the year 1948 and x 苷 50 represent the year 1950. b. Assume a new world record time will be established in 2008. Use the function you chose in a. to predict the world record time in the men’s 400-meter race for the year 2008. Round to the nearest hundredth of a second. 25.

TELECOMMUTING The graph below shows the projected growth in the number of telecommuters. Projected Growth in Telecommuting Number of telecommuters (in millions)

22.

531

Modeling Data with Exponential and Logarithmic Functions

12

9.6

10.4

11

11.2 11.4

9 6 3 0 1998 2000 2002 2004 2006

Which type of model, an increasing exponential model or an increasing logarithmic model, is more likely to provide a better fit for the data? Explain.

532 26.

Chapter 7

Exponential and Logarithmic Functions

POPULATION OF HAWAII The following table shows the population of the state of Hawaii for selected years from 1950 to 2004.

28.

CHEMISTRY The amount of oxygen x, in milliliters per liter, that can be absorbed by water at a certain temperature T, in degrees Fahrenheit, is given in the following table.

Population of the State of Hawaii Year

Population, P

Year

Population, P

1950

499,000

1985

1,039,698

1955

529,000

1990

1,113,491

1960

642,000

1995

1,196,854

1965

704,000

2000

1,212,670

1970

762,920

2001

1,227,024

1975

875,052

2002

1,244,898

1980

967,710

2004

1,262,840

29.

OPTOMETRY The near point p of a person is the closest distance at which the person can see an object distinctly. As one grows older, one’s near point increases. The table below shows data for the average near point of various people with normal eyesight. Near point p (cm)

15

11

20

13

25

15

30

17

35

20

40

23

50

26

a. Find an exponential regression model for these data. Round each constant to the nearest thousandth. b. What near point does this equation predict for a person 60 years old? Round to the nearest centimeter.

10.5

38

8.4

46

7.6

52

7.1

58

6.8

64

6.5

THE HENDERSON-HASSELBACH FUNCTION The scientists Henderson and Hasselbach determined that the pH of blood is a function of the ratio q of the amounts of bicarbonate and carbonic acid in the blood. a. Determine a linear model and a logarithmic model for the data. Use q as the independent variable (domain) and pH as the dependent variable (range). State the correlation coefficient for each model. Round a and b to five decimal places and r to six decimal places. Which model provides the better fit for the data?

c. What is the carrying capacity of the model? Round to the nearest thousand.

Age y, years

32

b. Using your model, how much oxygen, to the nearest tenth of a milliliter per liter, can be absorbed in water that is 50 F?

a. Find a logistic growth model that approximates the population of the state of Hawaii as a function of the year. Use t 苷 0 to represent the year 1950.

27.

Oxygen absorbed, ml兾L

a. Find a logarithmic regression equation for these data. Round each constant to the nearest thousandth.

Source: economagic.com, http://www.economagic.com/ em-cgi/data.exe/beapi/a15300.

b. Use the model from a. to predict the population of the state of Hawaii for the year 2010. Round to the nearest ten thousand.

Temperature, F

q

7.9

12.6

31.6

50.1

79.4

pH

7.0

7.2

7.6

7.8

8.0

b. Use the model you chose in a. to find the q-value associated with a pH of 8.2. Round to the nearest tenth. 30.

WORLD POPULATION The following table lists the years in which the world’s population first reached 3, 4, 5, and 6 billion. World Population Milestones

Year

Population

1960

3 billion

1974

4 billion

1987

5 billion

1999

6 billion

Source: The World Almanac 2006, p. 851.

7.6

c. Use the model you selected in b. to predict the women’s Olympic gold medal high jump height in 2012. Round to the nearest tenth of an inch.

a. Find an exponential model for the data in the table. Let x 苷 0 represent the year 1960. b. Use the model to predict the year in which the world’s population will first reach 8 billion.

33.

PANDA POPULATION One estimate gives the world panda population as 3200 in 1980 and

31.

NUMBER OF CINEMA SITES The following table shows the number of U.S. indoor cinema sites for the years 1999 to 2004.

590 in 2000. a. Find an exponential model for the data and use the model to predict the year in which the panda population p will be reduced to 200. (Let t 苷 0 represent the year 1980.) b.

32.

533

Modeling Data with Exponential and Logarithmic Functions

Because the exponential model in a. fits the data perfectly, does this mean that the model will accurately predict future panda populations? Explain.

OLYMPIC HIGH JUMP The following table shows the Olympic gold medal heights for the women’s high jump from 1968 to 2004.

Year

Number of Indoor Cinema Sites, S

1999

7031

2000

6550

2001

5813

2002

5712

2003

5700

2004

5629

Source: National Association of Theatre Owners

a. Determine an exponential regression model and a logarithmic regression model for the data. Use t 苷 1 to represent the year 1999. Round the constants a and b to the nearest hundred thousandth. State the correlation coefficient r for each model. b. Examine the correlation coefficients to determine which model provides a better fit for the data.

Women’s Olympic High Jump, 1968 to 2004 Year

Height

Year

1968

5 ft 11.75 in.

1988

c. Use the model you selected in b. to predict the number of U.S. indoor cinema sites for the year 2009.

Height

6 ft 8 in. 34.

1972

6 ft 3.65 in.

1992

6 ft 7.5 in.

1976

6 ft 4 in.

1996

6 ft 8.75 in.

1980

6 ft 5.5 in

2000

6 ft 7 in.

1984

6 ft 7.5 in.

2004

6 ft 9.1 in.

Source: Time Almanac 2006.

a. Determine a linear model and a logarithmic model for the data, with the height measured in inches. State the correlation coefficient r for each model. Represent the year 1968 by x 苷 1, the year 1972 by x 苷 2, and the year 2004 by x 苷 10. b. Examine the correlation coefficients to determine which model provides a better fit for the data.

TEMPERATURE OF COFFEE A cup of coffee is placed in a room that maintains a constant temperature of 70°F. The following table shows both the coffee temperature T after t minutes and the difference between the coffee temperature and the room temperature after t minutes.

Time t (minutes)

0

5

10

15

20

25

Coffee temp. T ( F)

165

140

121

107

97

89

T 70

195

170

151

137

27

19

a. Find an exponential model for the difference T 70 as a function of t. b. Use the model in a. to predict how long it will take (to the nearest minute) for the coffee to cool to 80°F.

534 35.

Chapter 7

Exponential and Logarithmic Functions b. According to the logistic growth model, what will the world’s population approach as t l ? Round to the nearest billion.

WORLD POPULATION The following table lists the years in which the world’s population first reached 3, 4, 5, and 6 billion. World Population Milestones Year

Population

1960

3 billion

1974

4 billion

1987

5 billion

1999

6 billion

36. A CORRELATION COEFFICIENT OF 1 A scientist uses a graphing calculator to model the data set 兵共2, 5兲, 共4, 6兲其 with a logarithmic function. The following display shows the results. LnReg y=a+blnx a=4 b=1.442695041 r2=1 r=1

Source: The World Almanac 2006, p. 851.

a. Find a logistic growth model P共t兲 for the data in the table. Let t represent the number of years after 1960 (t 苷 0 represents the year 1960).

What is the significance of the fact that the correlation coefficient for the regression equation is r 苷 1?

Connecting Concepts 37.

A 苷兵共2, 5兲, 共3, 10兲, 共4, 17兲, 共4, 17兲, 共5, 28兲其

Some data sets can best be modeled by a power function. On a TI-83/TI-83 Plus/TI-84 Plus, the PwrReg instruction is used to produce a power regression function for a set of data.

Because the ordered pair 共4, 17兲 is listed twice, the engineer decides to eliminate one of these ordered pairs and model the data in set B.

a. Find an exponential regression function and a power regression function for the following data. State the correlation coefficient r for each model.

DUPLICATE DATA POINTS An engineer needs to model the data in set A with an exponential function.

B 苷兵共2, 5兲, 共3, 10兲, 共4, 17兲, 共5, 28兲其 Determine whether A and B both have the same exponential regression function. 38.

DOMAIN ERROR A scientist needs to model the data in set A.

a. When the scientist attempts to use a graphing calculator to determine the logarithmic regression equation, the calculator displays the message “ERR:DOMAIN”

1

2

3

4

5

6

y

2.1

5.5

9.8

14.6

20.1

25.8

b. Which of the two regression functions provides the better fit for the data?

A 苷兵共0, 1.2兲, 共1, 2.3兲, 共2, 2.8兲, 共3, 3.1兲, 共4, 3.3兲, 共5, 3.4兲其 The scientist views a scatter plot of the data and decides to model the data with a logarithmic function of the form y 苷 a b ln x.

x

40.

PERIOD OF A PENDULUM The following table shows the time t (in seconds) of the period of a pendulum of length l (in feet). (Note: The period of a pendulum is the time it takes the pendulum to complete a swing from the right to the left and back.)

Length l Time t

1

2

3

4

6

8

1.11

1.57

1.92

2.25

2.72

3.14

Explain why the calculator was unable to determine the logarithmic regression equation for the data. b. Explain what the scientist could do so that the data in set A could be modeled by a logarithmic function of the form y 苷 a b ln x. 39.

POWER FUNCTIONS A function that can be written in the form y 苷 ax b is said to be a power function.

a. Determine the equation of the best model for the data. Your model must be a power function or an exponential function. b. According to the model you chose in a. what is the length of a pendulum, to the nearest tenth of a foot, that has a period of 12 seconds?

535

Exploring Concepts with Technology

Projects c. Use the three-step modeling process to find a regression equation that models the data.

A MODELING PROJECT The purpose of this Project is for you to find data that can be modeled by an exponential or a logarithmic function. Choose data from a real-life situation that you find interesting. Search for the data in a magazine, in a newspaper, in an almanac, or on the Internet. If you wish, you can collect your data by performing an experiment.

d. Graph the regression equation on the scatter plot of the data. What is the correlation coefficient for the model? Do you think your regression equation accurately models your data? Explain. e. Use the regression equation to predict the value of

a. List the source of your data. Include the date, page number, and any other specifics about the source. If your data were collected by performing an experiment, then provide all the details of the experiment. b. Explain what you have chosen as your variables. Which variable is the dependent variable and which variable is the independent variable?

■

the dependent variable for a specific value of the independent variable

■

the independent variable for a specific value of the dependent variable Write a few comments about what you have learned from this Project.

f.

Exploring Concepts with Technology Using a Semilog Graph to Model Exponential Decay

Table 7.16 T

V

90

700

100

500

110

350

120

250

130

190

140

150

150

120

Consider the data in Table 7.16, which shows the viscosity V of SAE 40 motor oil at various temperatures T in degrees Fahrenheit. The graph of these data is shown below, along with a curve that passes through the points. The graph in Figure 7.47 appears to have the shape of an exponential decay model. One way to determine whether the graph in Figure 7.47 is the graph of an exponential function is to plot the data on semilog graph paper. On this graph paper, the horizontal axis remains the same, but the vertical axis uses a logarithmic scale. The data in Table 7.16 are graphed again in Figure 7.48, but this time the vertical axis is a natural logarithm axis. This graph is approximately a straight line.

V Natural logarithm of viscosity

V 700 600 Viscosity

1.

500 400 300 200 100 90

100

110

120

130

140

150

T

1n 700 1n 500 1n 300

1n 100 90

100

110

120

130

140

Temperature, in °F

Temperature, in °F

Figure 7.47

Figure 7.48

The slope of the line in Figure 7.48, to the nearest ten-thousandth, is m苷

ln 500 ln 120 ⬇ 0.0285 100 150

150

T

536

Chapter 7

Exponential and Logarithmic Functions Using this slope and the point-slope formula with V replaced by ln V, we have ln V ln 120 苷 0.0285共T 150兲 ln V ⬇ 0.0285T 9.062 (1) Equation (1) is the equation of the line on a semilog coordinate grid. Now solve Equation (1) for V. e ln V 苷 e0.0285T9.062 V 苷 e0.0285Te 9.062 (2) V ⬇ 8621e0.0285T Equation (2) is a model of the data in the rectangular coordinate system shown in Figure 7.47.

1. A chemist wishes to determine the decay characteristics of iodine-131. A 100-milligram sample of iodine-131 is observed over a 30-day period. Table 7.17 shows the amount A (in milligrams) of iodine-131 remaining after t days.

Table 7.17 t

A

1

91.77

4

70.92

8

50.30

15

27.57

20

17.95

30

7.60

a. Graph the ordered pairs 共t, A兲 on semilog paper. (Note: Semilog paper comes in different varieties. Our calculations are based on semilog paper that has a natural logarithm scale on the vertical axis.) b. Use the points 共4, 4.3兲 and 共15, 3.3兲 to approximate the slope of the line that passes through the points. c. Using the slope calculated in b. and the point 共4, 4.3兲, determine the equation of the line. d. Solve the equation you derived in c. for A. e. Graph the equation you derived in d. in a rectangular coordinate system. f.

2. The birth rates B per thousand people in the United States are given in Table 7.18 for the years 1986 through 1990 (t 苷 0 corresponds to 1986).

Table 7.18 t

B

0

15.5

1

15.7

2

15.9

3

16.2

4

16.7

a. Graph the ordered pairs 共t, ln B兲. (You will need to adjust the scale so that you can discriminate between plotted points. A suggestion is given in Figure 7.49.) b. Use the points 共1, 2.754兲 and 共3, 2.785兲 to approximate the slope of the line that passes through the points. c. Using the slope calculated in b. and the point 共1, 2.754兲, determine the equation of the line.

1n B Natural logarithm of birth rate

What is the half-life of iodine-131?

d. Solve the equation you derived in c. for B.

2.81 2.80

e. Graph the equation you derived in d. in a rectangular coordinate system.

2.79 2.78

f.

2.77 2.76

If the birth rate continues as predicted by your model, in what year will the birth rate be 17.5 per thousand?

2.75 2.74 1

2

3

Year (t = 0 is 1986)

Figure 7.49

4

t

The difference in graphing strategies between Exercise 1 and Exercise 2 is that in Exercise 1, semilog paper was used. When a point is graphed on this coordinate paper, the y-coordinate is ln y. In Exercise 2, graphing a point 共x, ln y兲 in a rectangular coordinate system has the same effect as graphing 共x, y兲 in a semilog coordinate system.

Summary

537

Chapter 7 Summary 7.1 Exponential Functions and Their Applications • For all positive real numbers b, b 苷 1, the exponential function defined by f 共x兲苷 bx has the following properties: 1. f has the set of real numbers as its domain.

7.3 Properties of Logarithms and Logarithmic Scales • If b, M, and N are positive real numbers 共b 苷 1兲, and p is any real number, then

2. f has the set of positive real numbers as its range.

log b共MN 兲苷 log b M log b N

3. f has a graph with a y-intercept of 共0, 1兲.

log b

4. f has a graph that is asymptotic to the x-axis. 5. f is a one-to-one function. 6. f is an increasing function if b 1. 7. f is a decreasing function if 0 b 1. • As n increases without bound, 共1 1兾n兲n approaches an irrational number denoted by e. The value of e accurate to eight decimal places is 2.71828183. • The function defined by f 共x兲苷 e x is called the natural exponential function.

7.2 Logarithmic Functions and Their Applications • Definition of a Logarithm If x 0 and b is a positive constant 共b 苷 1兲, then y 苷 log b x

if and only if b y 苷 x

• For all positive real numbers b, b 苷 1, the function defined by f 共x兲苷 log b x has the following properties: 1. f has the set of positive real numbers as its domain. 2. f has the set of real numbers as its range. 3. f has a graph with an x-intercept of 共1, 0兲. 4. f has a graph that is asymptotic to the y-axis. 5. f is a one-to-one function. 6. f is an increasing function if b 1. 7. f is a decreasing function if 0 b 1. • The exponential form of y 苷 log b x is b y 苷 x.

M 苷 log b M log b N N

log b共M p 兲苷 p log b M log b M 苷 log b N implies M 苷 N M 苷 N implies

log b M 苷 log b N

• Change-of-Base Formula If x, a, and b are positive real numbers with a 苷 1 and b 苷 1, then log b x 苷

log a x log a b

冉冊

• An earthquake with an intensity of I has a Richter scale I magnitude of M 苷 log , where I0 is the measure of the I0 intensity of a zero-level earthquake. • The pH of a solution with a hydronium-ion concentration of H mole per liter is given by pH 苷 log关H 兴.

7.4 Exponential and Logarithmic Equations • Equality of Exponents Theorem If b is a positive real number 共b 苷 1兲 such that b x 苷 b y, then x 苷 y. • Exponential equations of the form b x 苷 b y can be solved by using the Equality of Exponents Theorem. • Exponential equations of the form b x 苷 c can be solved by taking either the common logarithm or the natural logarithm of each side of the equation. • Logarithmic equations can often be solved by using the properties of logarithms and the definition of a logarithm.

• The logarithmic form of b y 苷 x is y 苷 log b x.

7.5 Exponential Growth and Decay

• Basic Logarithmic Properties 1. log b b 苷 1

2. log b 1 苷 0

3. log b 共bx 兲苷 x

4. blogb x 苷 x

• The function f 共x兲苷 log10 x is the common logarithmic function. It is customarily written as f 共x兲苷 log x. • The function f 共x兲苷 log e x is the natural logarithmic function. It is customarily written as f 共x兲苷 ln x.

• The function defined by N共t兲苷 N0 e kt is called an exponential growth function if k is a positive constant, and it is called an exponential decay function if k is a negative constant. • The Compound Interest Formula A principal P invested at an annual interest rate r, expressed as a decimal and compounded n times per year for t years, produces the balance

冉冊

A苷P 1

r n

nt

538

Chapter 7

Exponential and Logarithmic Functions

• Continuous Compounding Interest Formula If an account with principal P and annual interest rate r is compounded continuously for t years, then the balance is A 苷 Pert. • The Logistic Model The magnitude of a population at time t is given by P共t兲苷

c 1 aebt

where P0 苷 P共0兲 is the population at time t 苷 0, c is the carrying capacity of the population, and b is a constant called the growth rate constant.

7.6 Modeling Data with Exponential and Logarithmic Functions

• If the graph of f lies below all of its tangents on 关x1 , x2 兴, then f is concave downward on 关x1 , x2 兴. • The Modeling Process Use a graphing utility to 1. construct a scatter plot of the data to determine which type of function will best model the data. 2. find the equation of the modeling function and the correlation coefficient or the coefficient of determination for the equation. 3. examine the correlation coefficient or the coefficient of determination and view a graph that displays both the function and the scatter plot to determine how well the function fits the data.

• If the graph of f lies above all of its tangents on 关x1 , x2 兴, then f is concave upward on 关x1 , x2 兴.

Chapter 7 Assessing Concepts In Exercises 1 to 5, determine whether the statement is true or false. If the statement is false, give an example or state a reason to demonstrate that the statement is false.

11. Every function has an inverse function. 12. If 7 x 苷 40, then log 7 40 苷 x. 13. If log 4 x 苷 3.1, then 43.1 苷 x. 14. The exponential function h共x兲苷 b x is an increasing function. 15. The logarithmic function j共x兲苷 log b x is an increasing function.

In Exercises 6 to 12, match an expression to each description. A letter may be used more than once. Some letters may not be needed.

16. A function which is symmetric with respect to the y-axis. ______

a. f (x) 苷 ln(x 4) b. f(x) 苷

17. An increasing function which is symmetric with respect to the c. origin. _____

f (x) 苷

ex ex 2 ex ex 2

18. A logistic function. ______

d. f (x) 苷 log 1/2 x

19. An increasing function with a vertical asymptote. ______

e. f (x) 苷

10. A decreasing function with a horizonal asymptote. _____

f.

11. A decreasing function with a vertical asymptote. ______ 12. An increasing function which is concave upward. ______

f (x) 苷 x0

1 x e 2 485 , 1 9.5 e0.019x

g. f (x) 苷 e x

539

Review Exercises

Chapter 7 Review Exercises In Exercises 1 to 12, solve each equation. Do not use a calculator.

In Exercises 33 to 36, expand the given logarithmic expression.

11. log 5 25 苷 x

12. log 3 81 苷 x

13. ln e 3 苷 x

33. log b

14. ln ep 苷 x

15. 32x7 苷 27

16. 5x4 苷 625

1 8

18. 27共3x 兲苷 31

19. log x 2 苷 6

1 log 兩x兩苷 5 2

11. 10 log 2x 苷 14

12. eln x 苷 64

17. 2x 苷

10.

2

x 2y 3 z

34. log b

35. ln xy 3

36. ln

兹x y 2z

兹xy z4

In Exercises 37 to 40, write each logarithmic expression as a single logarithm with a coefficient of 1.

37. 2 log x

1 log共x 1兲 3

38. 5 log x 2 log共x 5兲

In Exercises 13 to 22, sketch the graph of each function.

冉冊 1 4

13. f 共x兲苷共2.5兲x

14. f 共x兲苷

15. f 共x兲苷 3兩x兩

16. f 共x兲苷 4兩x兩

17. f 共x兲苷 2 3 x

1 log x 3

19. f 共x兲苷

1 21. f 共x兲苷 ln x 2

x

39.

1 ln 2xy 3 ln z 2

40. ln x 共ln y ln z兲

18. f 共x兲苷 2

In Exercises 41 to 44, use the change-of-base formula and a calculator to approximate each logarithm accurate to six significant digits.

20. f 共x兲苷 3 log x 1/3

41. log 5 101

42. log 3 40

43. log 4 0.85

44. log 8 0.3

共x3兲

22. f 共x兲苷 ln 兩x兩 In Exercises 45 to 60, solve each equation for x. Give exact answers. Do not use a calculator.

In Exercises 23 and 24, use a graphing utility to graph each function.

23. f 共x兲苷

4x 4x 2

24. f 共x兲苷

3x 3x 2

45. 4x 苷 30

46. 5x1 苷 41

47. ln 3x ln共x 1兲苷 ln 4

48. ln 3x ln 2 苷 1

49. e ln共x2兲苷 6

50. 10 log共2x1兲苷 31

4x 4x 苷2 4x 4x

52.

5x 5x 苷8 2

In Exercises 25 to 28, change each logarithmic equation to its exponential form.

51.

25. log 4 64 苷 3

26. log1/2 8 苷 3

53. log共log x兲苷 3

54. ln共ln x兲苷 2

27. log 兹2 4 苷 4

28. ln 1 苷 0

55. log 兹x 5 苷 3

56. log x log共x 15兲苷 1

57. log 4共log 3 x兲苷 1

58. log 7共log 5 x 2 兲苷 0

In Exercises 29 to 32, change each exponential equation to its logarithmic form.

59. log 5 x 3 苷 log 5 16x

60. 25 苷 16log

29. 53 苷 125

30. 210 苷 1024

31. 10 0 苷 1

32. 81/2 苷 2 兹2

61. EARTHQUAKE MAGNITUDE Determine, to the nearest 0.1, the Richter scale magnitude of an earthquake with an intensity of I 苷 51,782,000I0 .

4

x

540

Chapter 7

Exponential and Logarithmic Functions

62. EARTHQUAKE MAGNITUDE A seismogram has an amplitude of 18 millimeters and the difference in time between the s-wave and the p-wave is 21 seconds. Find, to the nearest tenth, the Richter scale magnitude of the earthquake that produced the seismogram.

c. How long, to the nearest day, will it take for 90% of the wound to heal? In Exercises 71 to 74, find the exponential growth or decay function N(t) N0 ekt that satisfies the given conditions.

63. COMPARISON OF EARTHQUAKES An earthquake had a Richter scale magnitude of 7.2. Its aftershock had a Richter scale magnitude of 3.7. Compare the intensity of the earthquake to the intensity of the aftershock by finding, to the nearest unit, the ratio of the larger intensity to the smaller intensity.

71. N共0兲苷 1, N共2兲苷 5

64. COMPARISON OF EARTHQUAKES An earthquake has an intensity 600 times the intensity of a second earthquake. Find, to the nearest tenth, the difference between the Richter scale magnitudes of the earthquakes.

74. N共1兲苷 2, N共0兲苷 1

72. N共0兲苷 2, N共3兲苷 11 73. N共1兲苷 4, N共5兲苷 5

75. POPULATION GROWTH a. Find the exponential growth function for a city whose population was 25,200 in 2005 and 26,800 in 2006. Use t 苷 0 to represent the year 2005.

65. CHEMISTRY Find the pH of tomatoes that have a hydronium-ion concentration of 6.28 10 5. Round to the nearest tenth.

b. Use the growth function to predict, to the nearest hundred, the population of the city in 2012.

66. CHEMISTRY Find the hydronium-ion concentration of rainwater that has a pH of 5.4. 67. COMPOUND INTEREST Find the balance when $16,000 is invested at an annual rate of 8% for 3 years if the interest is compounded a. monthly b. continuously 68. COMPOUND INTEREST Find the balance when $19,000 is invested at an annual rate of 6% for 5 years if the interest is compounded a. daily b. continuously

76. CARBON DATING Determine, to the nearest ten years, the age of a bone if it now contains 96% of its original amount of carbon-14. The half-life of carbon-14 is 5730 years. 77.

CELLULAR TELEPHONE SUBSCRIBERSHIP The following table shows the number of U.S. cellular telephone subscriptions, in thousands, for selected years from 1990 to 2004.

Year

Number of Cellular Telephone Subscriptions (in thousands)

1990

5283

69. DEPRECIATION The scrap value S of a product with an expected life span of n years is given by S共n兲苷 P共1 r兲n, where P is the original purchase price of the product and r is the annual rate of depreciation. A taxicab is purchased for $12,400 and is expected to last 3 years. What is its scrap value if it depreciates at a rate of 29% per year?

1992

11,033

1994

24,134

1996

44,043

70. MEDICINE A skin wound heals according to the function given by N共t兲苷 N0 e 0.12t, where N is the number of square centimeters of unhealed skin t days after the injury, and N0 is the number of square centimeters covered by the original wound. a. What percentage of the wound will be healed after 10 days? b. How many days, to the nearest day, will it take for 50% of the wound to heal?

1998

69,209

2000

109,478

2002

140,767

2004

182,140

Source: The World Almanac 2006

Find the equation of the mathematical model that you believe will most accurately predict the number of U.S. cellular telephone subscriptions for the year 2008. Explain the reasoning you used to select your model.

541

Quantitative Reasoning 78.

b. Examine the correlation coefficients of the regression models to determine which model provides the better fit.

MORTALITY RATE The following table shows the infant mortality rate in the United States for selected years from 1960 to 2003. (Source: The World Almanac 2006.)

c. Use the model you selected in b. to predict, to the nearest 0.1, the infant mortality rate in 2008.

U.S. Infant Mortality Rate, 1960 –2003 (per 1000 live births) Year

Rate, R

1960

26.0

1970

20.0

1980

12.6

1990

9.2

1995

7.6

2000

6.9

2001

6.8

2002

7.0

2003

6.9

79. LOGISTIC GROWTH The population of coyotes in a national park satisfies the logistic model with P0 苷 210 in 1997, c 苷 1400, and P共3兲苷 360 (the population in 2000). a. Determine the logistic model. b. Use the model to predict, to the nearest 10, the coyote population in 2010. 80. Consider the logistic function P共t兲苷

128 1 5e0.27t

a. Find P0 . b. What does P共t兲 approach as t l

?

a. Find an exponential model and a logarithmic model for the infant mortality rate, R, as a function of the year. Represent the year 1960 by t 苷 60.

»» »»»» Quantitative Reasoning: Sales »»» DIGITAL CAMERA SALES The following table shows the worldwide sales, in millions, of digital cameras for the years 1999 to 2003. (Source: Digital Photography Review) Year

1999

2000

2001

2002

2003

Worldwide Sales of Digital Cameras (in millions)

5.5

11.0

18.5

30.5

50.0

QR1. Find an exponential model and a logistic model for the data. Let t 苷 0 represent the year 1999. Round constants to the nearest hundred thousandth. QR2. Use each of the models to predict the number of sales of digital cameras for the year 2009. Round to the nearest tenth of a million.

542

Chapter 7

Exponential and Logarithmic Functions A business analyst thinks that the digital camera market has started to reach its saturation point. The analyst predicts that during the 2004 to 2007 period, the sales of digital cameras will be as shown in the following table. Year

2004

2005

2006

2007

Projected Worldwide Sales of Digital Cameras (in millions)

59.3

69.2

77.2

82.5

QR3. Find a logarithmic model for the data in the above table. Let t 苷 1 represent the year 2004. Round constants to the nearest hundred thousandth. QR4. Use the model from Exercise QR3 to predict the number of sales of digital cameras for the year 2009. Round to the nearest tenth of a million. QR5. Find a logistic model for all the data by combining the tables. Let t 苷 0 represent the year 1999 and t 苷 5 represent the year 2004. Use this model to predict the number of sales for the year 2009. QR6. The answers in Exercises QR2, QR4, and QR5 provide us with four predictions for the number of digital camera sales in the year 2009. Which of these predicted values do you think is the most realistic?

Chapter 7 Test 11. a. Write log b共5x 3兲苷 c in exponential form. b. Write 3x/2 苷 y in logarithmic form.

12. Expand log b

z2 y 兹x 3

.

13. Write log共2x 3兲 3 log共x 2兲 as a single logarithm with a coefficient of 1.

14. Use the change-of-base formula and a calculator to approximate log 4 12. Round your result to the nearest ten thousandth.

15. Graph: f 共x兲苷 3x/2 16. Graph: f 共x兲苷 ln共x 1兲 17. Solve: 5x 苷 22 . Round your solution to the nearest ten thousandth.

18. Find the exact solution of 45x 苷 7x. 19. Solve: log共x 99兲 log共3x 2兲苷 2

10. Solve: ln共2 x兲 ln共5 x兲苷 ln共37 x兲 11. Find the balance on $20,000 invested at an annual interest rate of 7.8% for 5 years, compounded a. monthly b. continuously 12. COMPOUND INTEREST Find the time required for money invested at an annual rate of 4% to double in value if the investment is compounded monthly. Round to the nearest hundredth of a year. 13. EARTHQUAKE MAGNITUDE a. What, to the nearest tenth, will an earthquake measure on the Richter scale if it has an intensity of I 苷 42,304,000I0 ? b. Compare the intensity of an earthquake that measures 6.3 on the Richter scale with the intensity of an earthquake that measures 4.5 on the Richter scale by finding the ratio of the larger intensity to the smaller intensity. Round to the nearest whole number.

Cumulative Review Exercises

543

14. a. Find the exponential growth function for a city whose population was 34,600 in 2001 and 39,800 in 2004. Use t 苷 0 to represent the year 2001.

a. Find a logarithmic model and a logistic model for the data. Use t 苷 1 to represent the year 1999 and t 苷 7 to represent the year 2005.

b. Use the growth function in a. to predict the population of the city in 2011. Round to the nearest thousand.

b. Use each of the models from a. to predict the women’s world record javelin throw distance for the year 2010. Round to the nearest hundredth of a meter.

15. Determine, to the nearest ten years, the age of a bone if it now contains 92% of its original amount of carbon-14. The half-life of carbon-14 is 5730 years. 16. a.

Find the exponential regression function for the following data. 兵共2.5, 16兲, 共3.7, 48兲, 共5.0, 155兲, 共6.5, 571兲, 共6.9, 896兲其

b. Use the function to predict, to the nearest whole number, the y value associated with x 苷 7.8. 17.

18. POPULATION GROWTH The population of raccoons in a state park satisfies a logistic growth model with P0 苷 160 in 2003 and P共1兲苷 190. A park ranger has estimated the carrying capacity of the park to be 1100 raccoons. a. Determine the logistic growth model for the raccoon population where t is the number of years after 2003. b. Use the logistic model from a. to predict the raccoon population in 2010.

WOMEN’S JAVELIN THROW The following table shows the progression of the world record distances for the women’s javelin throw from 1999 to 2005. (Source: http://www.athletix.org/Statistics/wrjtwomen.html) World Record Progression in the Women’s Javelin Throw Year

Distance in meters, d

1999

67.09

2000

68.22

2000

69.48

2001

71.54

2005

71.70

Cumulative Review Exercises 1. Given f 共x兲苷 cos x and g共x兲苷 x2 1, find 共 f ⴰ g兲共x兲.

4. For the right triangle shown at the right, find a. Round to the nearest centimeter.

2. Given f 共x兲苷 2x 8, find f 1共x兲. 3. For the right triangle shown at the right, find sin , cos , and tan .

a

26° 15 cm 3

θ 4

冉

冊

5. What are the amplitude, period, and phase shift of the graph of y 苷 4 cos 2x ? 2

544

Chapter 7

Exponential and Logarithmic Functions

6. What is the amplitude and period of the graph of y 苷 sin x cos x? 7. Is the sine function an even function, an odd function, or neither an even nor an odd function? 8. Verify the identity

1 sin x 苷 cos x cot x. sin x

冉冉冊冊

9. Evaluate tan sin1

12 13

.

10. Solve 2 cos x sin x 1 苷 0 for 0 x 2.

14. For triangle ABC, B 苷 32°, a 苷 42 feet, and b 苷 50 feet. Find the measure of angle A. Round to the nearest degree. 15. Use De Moivre’s Theorem to find 共1 i兲8. 16. Find the two square roots of the imaginary number i. 17. Transform the point 共1, 1兲 in a rectangular coordinate system to a point in a polar coordinate system. 18. Sketch the graph of the polar equation r 苷 2 2 cos .

2

11. Find the magnitude and direction angle for the vector 具3, 4典. Round the angle to the nearest tenth of a degree. 12. Find the angle between the vectors v 苷具2, 3典 and w 苷具3, 4典. Round to the nearest tenth of a degree. 13. GROUND SPEED AND COURSE OF A PLANE An airplane is traveling with an airspeed of 400 mph at a heading of 48°. A wind of 55 mph is blowing at a heading of 115°. Find the ground speed and the course of the plane.

19. Solve 5x 苷 10. Round to the nearest hundredth. 20. RADIOACTIVE DECAY The half-life of an isotope of polonium is 138 days. If an ore sample originally contained 3 milligrams of polonium, how many milligrams of polonium remain after 100 days? Round to the nearest tenth of a milligram.

Solutions to the Try Exercises Use a test value from each of the intervals

Exercise Set 1.1, page 12

冉

,

30s 66 24s 60 苷 0 6s 126 苷 0

2

2x 5 1 or

30. x x 2 苷 0

x苷

b苷1

3 2

1 兹1 8 1 3 苷苷 2 2

2x 5 1 2x 6

x2

x3

80. 兩3 2x兩 5

5 3 2x 5 8 2x 2

1 3 x苷苷 2 2

4

1

x

The solution set is 关1, 4兴.

The solutions are 1 and 2. 44. 12w 2 41w 24 苷 0

Exercise Set 1.2, page 27

共4w 3兲共3w 8兲苷 0

6. d 苷兹共x2 x1 兲2 共 y2 y1 兲2

4w 3 苷 0 or 3w 8 苷 0

The solutions are

d 苷兹关10 共5兲兴 2 共14 8兲2

8 w苷 3 8 3 and . 4 3

苷兹共5兲2 62 苷兹25 36 苷兹61 26.

2

3

3x 21 10x 40

3 x

−3

7x 61

The solution set is

y

30.

y

58. 3共x 7兲 5共2x 8兲

61 x 7

−6

冋冊

32.

40. y-intercept:

y

61 , . 7

and x 苷

3 2

0,

x-intercept: 共5, 0兲

5

−3

共6x 5兲共2x 3兲 0 5 6

冉冊 15 4

y

4

12x 2 8x 15 0

x苷

x

−2

−3

12x 2 8x 15

66.

5 , . 6

The solution set is 共, 2兴关3, 兲.

1 兹1 4共1兲共2兲 2共1兲 2

3 w苷 4

册冋冊

2x 4 c 苷 2

1 3 x苷苷 1 or 2

冉

78. 兩2x 5兩 1

2

a苷1

5 6

The solution set is ,

• Subtract 3x from each side. • Divide each side by 5.

0

−3

20. 3x 5y 苷 15

3 y苷 x3 5

冊冉冊

++ | − − − − − − − − | + ++

126 s苷苷 21 6 5y 苷 3x 15

冊冉

3 3 5 5 , , , and , to determine 2 2 6 6 2 where 12x 8x 15 is positive.

6. 6共5s 11兲 12共2s 5兲苷 0

3 x

−4

4

x

• Critical values

S1

S2

Solutions to the Try Exercises 54. a. 关0, 兲

64. r 苷兹共1 共2兲兲2 共7 5兲2

苷兹9 4 苷兹13

b. Since $31,250 is between $30,650 and $74,200 use

Using the standard form

T共x兲苷 0.25共x 30,650兲 4220. Then, T共31,250兲苷 0.25共31,250 30,650兲 4220 苷 $4370.

共x h兲2 共 y k兲2 苷 r 2

c. Since $78,900 is between $74,200 and $154,800, use

T共x兲苷 0.28共x 74,200兲 15,107.50 Then,

with h 苷 2, k 苷 5, and r 苷兹13 yields

T共78,900兲苷 0.28共78,900 74,200兲 15,107.50 苷 $16,423.50.

共x 2兲2 共 y 5兲2 苷共兹13 兲

2

66.

x 2 y 2 6x 4y 12 苷 0

56. a. This is the graph of a function. Every vertical line

x 2 6x y 2 4y 苷 12

intersects the graph in at most one point.

x 2 6x 9 y 2 4y 4 苷 12 9 4

b. This is not the graph of a function. Some vertical lines

共x 3兲2 共 y 2兲2 苷 12

intersect the graph at two points.

center 共3, 2兲, radius 1

c. This is not the graph of a function. The vertical line at

x 苷 2 intersects the graph at more than one point.

Exercise Set 1.3, page 46

d. This is the graph of a function. Every vertical line

intersects the graph at exactly one point.

2. Given g共x兲苷 2x 2 3 a. g共3兲苷 2共3兲2 3 苷 18 3 苷 21

72. v共t兲苷 44,000 4200t, 0 t 8

b. g共1兲苷 2共1兲 3 苷 2 3 苷 5

74. a. V共x兲苷共30 2x兲2x

2

苷共900 120x 4x 2 兲x

c. g共0兲苷 2共0兲2 3 苷 0 3 苷 3 d. g

冉冊冉冊 1 2

1 2

苷2

2

3苷

苷 900x 120x 2 4x 3

7 1 3苷 2 2

b. Domain: 兵x 兩 0 x 15其

e. g共c兲苷 2共c兲 3 苷 2c 3 2

2

78. AB 苷兹1 x2. The time required to swim from A to

f. g共c 5兲苷 2共c 5兲2 3 苷 2c 2 20c 50 3

B at 2 mph is

苷 2c 2 20c 53

BC 苷 3 x. The time required to run from B to C at 3x 8 mph is hours. 8

10. a. Because 0 0 5, Q共0兲苷 4. b. Because 6 a 7, Q共e兲苷 a 9. c. Because 1 n 2, Q共n兲苷 4.

Thus the total time to reach point C is

d. Because 1 m 2, 8 m2 7 11. Thus

Q共m2 7兲苷兹共m2 7兲 7 苷兹m2 苷 m 14. x 2 2y 苷 2

兹1 x 2 hours. 2

t苷

兹1 x 2 3 x hours 2 8

• Solve for y.

2y 苷 x 2 2

Exercise Set 1.4, page 64

1 y 苷 x2 1 2

14. a. The graph is symmetric with respect to the x-axis

because replacing y with y leaves the equation unaltered.

y is a function of x because each x value will yield one and only one y value.

b. The graph is not symmetric with respect to the y-axis

because replacing x with x alters the equation.

28. Domain is the set of all real numbers. 40. Domain is the set of all real numbers.

24. The graph is symmetric with respect to the origin be-

y

cause 共y兲苷共x兲3 共x兲 simplifies to y 苷 x 3 x, which is equivalent to the original equation y 苷 x 3 x.

4

−4

−2

4

x

44. Even, because h共x兲苷共x兲2 1 苷 x 2 1 苷 h共x兲. 58.

g(x − 3) g(x) − 2

68.

y

2

−4 −2 −2 −4

y 4

E(−x)

2 2

4 x

−4 −2 −2

2 −E(x)

4 x

S3

Solutions to the Try Exercises 70.

c. A is decreasing on 关0, 6兴 and on 关8, 10兴.

y 8

A is increasing on 关6, 8兴 and on 关10, 14兴. Area, in square inches

6 4 2 −2 −2

2

4 x

72. a.

y

−8

−6

−4

−2

y = g (2x)

2

4

A(t)

6 5 4 3 2 1 2 4 6 8 10 12 t t, in seconds

6

8

x

d. The highest point on the graph of A for 0 t 14

occurs when t 苷 0 seconds.

−2

72. a. On 关2, 3兴, b.

y

a苷2

1 y = g 2x

t 苷 3 2 苷 1

1 −8

−6

−4

−2

2

4

6

8

x

s共a兲苷 s共2兲苷 6 22 苷 24

−2

Average velocity 苷

Exercise Set 1.5, page 77 10. f 共x兲 g共x兲苷兹x 4 x

s共a t兲苷 s共3兲苷 6 32 苷 54

Domain: 兵x 兩 x 4其

苷

f 共x兲 g共x兲苷兹x 4 x Domain: 兵x 兩 x 4其 f 共x兲 g共x兲苷 x兹x 4 f 共x兲兹x 4 苷 g共x兲 x 14.

共 f g兲共x兲苷共x 2 3x 2兲共2x 4兲苷 x 2 x 2 共 f g兲共7兲苷共7兲2 共7兲 2 苷 49 7 2 苷 54

f 共x h兲 f 共x兲关4共x h兲 5兴共4x 5兲苷 30. h h

s共3兲 s共2兲 1

苷 54 24 苷 30 feet per second

Domain: 兵x 兩 x 4其 Domain: 兵x 兩 x 4其

s共a t兲 s共a兲 t

This is identical to the slope of the line through 共2, s共2兲兲 and 共3, s共3兲兲 because m苷

s共3兲 s共2兲苷 s共3兲 s共2兲苷 54 24 苷 30 32

b. On 关2, 2.5兴,

a苷2

苷

4x 4共h兲 5 4x 5 h

t 苷 2.5 2 苷 0.5

苷

4共h兲苷4 h

Average velocity 苷

s共2.5兲 s共2兲 37.5 24 苷 0.5 0.5

苷

13.5 苷 27 feet per second 0.5

s共a t兲苷 s共2.5兲苷 6共2.5兲2 苷 37.5

38. 共 g ⴰ f 兲共x兲苷 g关 f 共x兲兴苷 g关2x 7兴

苷 3关2x 7兴 2 苷 6x 19 共 f ⴰ g兲共x兲苷 f 关 g共x兲兴苷 f 关3x 2兴苷 2关3x 2兴 7 苷 6x 3 50. 共 f ⴰ g兲共4兲苷 f 关 g共4兲兴苷 f 关42 5共4兲兴

苷 f 关4兴苷 2共4兲 3 苷 5 66. a. l 苷 3 0.5t for 0 t 6, and l 苷 3 0.5t for t 6.

In either case, l 苷兩3 0.5t兩. The width is w 苷兩2 0.2t兩 as in Example 7.

b. A共t兲苷兩3 0.5t兩兩2 0.2t兩

c. On 关2, 2.1兴,

a苷2 t 苷 2.1 2 苷 0.1 s共a t兲苷 s共2.1兲苷 6共2.1兲2 苷 26.46 Average velocity 苷苷

s共2.1兲 s共2兲 26.46 24 苷 0.1 0.1 2.46 苷 24.6 feet per second 0.1

Continued

䉴

S4

Solutions to the Try Exercises d. On 关2, 2.01兴,

20. Check to see if f 关 g共x兲兴苷 x for all x in the domain of g

and g关 f 共x兲兴苷 x for all x in the domain of f. The following shows that f 关 g共x兲兴苷 x for all real numbers x.

a苷2 t 苷 2.01 2 苷 0.01

f 关 g共x兲兴苷 f 关2x 3兴

s共a t兲苷 s共2.01兲苷 6共2.01兲2 苷 24.2406

1 3 共2x 3兲 2 2

s共2.01兲 s共2兲 0.01

苷

苷

24.2406 24 0.01

苷x

苷

0.2406 苷 24.06 feet per second 0.01

Average velocity 苷

The following shows that g关 f 共x兲兴苷 x for all real numbers x.

冋冉

e. On 关2, 2.001兴,

g关 f 共x兲兴苷 g

a苷2

苷2

s共a t兲苷 s共2.001兲苷 6共2.001兲苷 24.024006 2

Average velocity 苷

s共2.001兲 s共2兲 0.001

苷

24.024006 24 0.001

32.

0.024006 苷 24.006 feet per second 0.001

f 共x兲苷 4x 8 y 苷 4x 8

• Replace f (x) by y.

x 苷 4y 8

• Interchange x and y.

x 8 苷 4y

苷

6共4 4共t兲共t兲2 兲 24 t

苷

24 24共t兲 6共t兲 24 t

苷

24t 6共t兲2 苷 24 6共t兲 t

y苷

1 x2 4

f 1共x兲苷

1 x2 4

2

38.

As t approaches zero, the average velocity approaches 24 feet per second.

f 共x兲苷

x ,x苷2 x2

y苷

x x2

• Replace f (x) by y.

x苷

y y2

• Interchange x and y.

xy 2x 苷 y

10. Because the graph of the given function is a line that

y共x 1兲苷 2x y苷 f 1共x兲苷

y

44.

(3, 2) 4

(6, 0) x

• Solve for y.

xy y 苷 2x

passes through 共0, 6兲, 共2, 3兲, and 共6, 3兲, the graph of the inverse will be a line that passes through 共6, 0兲, 共3, 2兲, and 共3, 6兲. See the following figure. Notice that the line shown below is a reflection of the line given in Exercise 10 across the line given by y 苷 x. Yes, the inverse relation is a function.

−4

• Replace y by f –1(x).

x共 y 2兲苷 y

Exercise Set 1.6, page 90

−2

• Solve for y.

1 共x 8兲苷 y 4

s共2 t兲 s共2兲 6共2 t兲2 24 苷 t t

2

册冊

Thus f and g are inverses.

f. On 关2, 2 t兴,

(−3, 6)

1 3 x 2 2

1 3 x 3 2 2 苷x33苷x

t 苷 2.001 2 苷 0.001

苷

3 3 苷x 2 2

2x x1 2x ,x苷1 x1

• Replace y by f –1(x) and indicate any restrictions.

f 共x兲苷兹4 x, x 4 y 苷兹4 x

• Replace f (x) by y.

x 苷兹4 y

• Interchange x and y.

x 苷4y 2

• Solve for y.

S5

Solutions to the Try Exercises

冉

x 2 4 苷 y

32. 45 苷 45

x 4 苷 y 2

1

f 共x兲苷 x 4, x 0 2

f –1(x)

• Replace y by and indicate any restrictions.

The range of f is 兵 y 兩 y 0其. Therefore, the domain of f is 兵x 兩 x 0其, as indicated above. 54.

• Replace K(x) by y.

x 苷 1.3y 4.7

• Interchange x and y.

x 4.7 苷 1.3y

• Solve for y.

x 4.7 苷y 1.3 K 1共x兲苷

x 4.7 1.3

冉

68. s 苷 r 苷 5 144

苷 4 ⬇ 12.57 meters 180

72. Let 2 be the angle through which the pulley with a di-

ameter of 0.8 meter turns. Let 1 be the angle through which the pulley with a diameter of 1.2 meters turns. Let r2 苷 0.4 meter be the radius of the smaller pulley, and let r1 苷 0.6 meter be the radius of the larger pulley.

1 苷 240 苷 • Replace y by K–1(x).

x 4.7 can be used to convert a 1.3 United Kingdom men’s shoe size to its equivalent U.S. shoe size.

18. a. Enter the data in the table. Then use your calculator

to find the linear regression equation: y 苷 3.410344828x 65.09359606 b. Evaluate the linear regression equation when x 苷 58.

y 苷 3.410344828共58兲 65.09359606 ⬇ 263

2 苷

0.6 0.4

b. Evaluate the quadratic regression model when

x 苷 40.

y 苷 0.05208共40 2 兲 3.5602共40兲 82.3299 ⬇ 23 The bird will consume approximately 23 milliliters of oxygen per minute.

Exercise Set 2.1, page 130 2. The measure of the complement of an angle of 87 is

共90 87 兲苷 3 The measure of the supplement of an angle of 87 is 共180 87 兲苷 93 14. Because 765 苷 2 360 45 , ⬔ is coterminal with

an angle that has a measure of 45 . ⬔ is a Quadrant I angle.

4 3

4 苷 2 radians or 360 3

74. The earth makes one revolution 共苷 2兲 in 1 day.

t 苷 24 3600 苷 86,400 seconds

苷

2 苷 ⬇ 7.27 10 5 radian/second t 86,400

80. C 苷 2r 苷 2共18 inches兲苷 36 inches

Thus one conversion factor is 共36p inches/1 revolution兲.

冉

冊

500 revolutions 500 revolutions 36p inches 苷 1 minute 1 minute 1 revolution 苷

32. a. Enter the data in the table. Then use your calculator

y 苷 0.05208x 2 3.56026x 82.32999

冉冊冉冊

0.42 苷 0.6

The ball will travel approximately 263 feet. to find the quadratic regression model:

4 radians 3

Thus r2 2 苷 r1 1

The function K 1共x兲苷

Exercise Set 1.7, page 103

冊

p p 180 苷 45 radian 苷 radian 4 4 p radians

1

K共x兲苷 1.3x 4.7 y 苷 1.3x 4.7

44.

冊冉冊

radians 苷 radian 180 4

18,000p inches 1 minute

Now convert inches to miles and minutes to hours. 18,000 inches 1 minute 苷

冉

冊冉

18,000 inches 1 foot 1 minute 12 inches

冊冉

1 mile 5280 feet

冊

60 minutes 1 hour

⬇ 54 miles per hour

Exercise Set 2.2, page 142 6. adj 苷兹82 52

adj 苷兹64 25 苷兹39 opp 5 sin u 苷苷 hyp 8 adj 兹39 苷 hyp 8

csc u 苷

hyp 8 苷 opp 5

hyp 8 8兹39 苷苷 adj 39 兹39 opp 5 5兹39 adj 兹39 tan u 苷苷苷 cot u 苷苷 adj 39 opp 5 兹39 cos u 苷

sec u 苷

S6

Solutions to the Try Exercises 4 opp 苷 , let opp 苷 4 and adj 苷 3. adj 3

18. Because tan u 苷

hyp 苷兹32 42 苷 5 sec u 苷 34. sin

兹3 兹2 cos tan 苷 1 3 4 4 2 2 兹6 4 兹6 1苷苷 4 4 h 116 h 苷 116 tan 68.9

tan 68.9 苷

h ⬇ 301 meters (three significant digits)

h

68.9°

58.

80 ft d

r 2兹3 苷 3 x

30. sec 苷

兹13 3

sec 苷

兹13 2

cot 苷

2 3

• Let r 2兹3 and x 3.

y 苷兹共2兹3兲2 32 苷兹3 y 苷兹3 because y 0 in Quadrant IV. 兹3

sin 苷

2兹3

苷

1 2

Thus cos 300 苷 cos 60 苷

1 . 2

Exercise Set 2.4, page 166

80 sin 5

10. t 苷

miles 9 miles 5280 feet 1 hour 苷 hour 1 hour 1 mile 60 minutes

9共5280兲 feet feet 苷苷 792 60 minute minute d r

917.9 feet ⬇ 1.2 minutes (to the nearest 792 feet per minute tenth of a minute) t⬇

66. 22° d 80

240 ft

h tan 22 苷 240 h 苷 240 tan 22 d 苷 80 h d 苷 80 240 tan 22 d ⬇ 180 feet 共two significant digits兲

y 苷 sin t

冉冊

苷 cos 苷 cos 苷

冉冊

7 4

苷 sin

4

苷 sin

兹2 2

冉冊冉

W

冊

苷

7 4

4

兹2 2

7 兹2 兹2 , 苷 4 2 2

16. The reference angle for

冉冊

sec

h

80 ft

7 ; W共t兲苷 P共x, y兲 where 4

x 苷 cos t

Change 9 miles per hour to feet per minute.

t苷

9 3 y 苷苷 x 6 2

80 d

d ⬇ 917.9 feet

r苷9

tan 苷

csc 苷

50. cos 300 0, 苷 360 300 苷 60

5°

d苷

y 9 3 3兹13 苷苷苷 r 13 3兹13 兹13 x 6 2 2兹13 苷苷 cos 苷苷 r 13 3兹13 兹13

38. 苷 255 180 苷 75

116 m

sin 5 苷

6. x 苷 6, y 苷 9, r 苷兹共6兲2 共9兲2 苷兹117 苷 3兹13

sin 苷

hyp 5 苷 adj 3

56.

Exercise Set 2.3, page 153

5 苷 sec 6 6 苷

5 is . 6 6

• sec t 0 for t in Quadrant III

2兹3 3

44. F共x兲苷 tan共x兲 sin共x兲

苷 tan x sin x 苷共tan x sin x兲

• tan x and sin x are odd functions.

苷 F共x兲 Because F共x兲苷 F共x兲, the function defined by F共x兲苷 tan x sin x is an odd function.

S7

Solutions to the Try Exercises 56.

y

32. y 苷 sin

P1(x, y)

3 x 4

y

a苷1

t x A(1, 0)

t−π

period 苷

P2(− x, − y)

8 3

1

2 8 2 苷苷 b 3兾4 3

34. y 苷 cos 3x

y tan t 苷 x tan共t 兲苷

• From the unit circle

2 2 2 period 苷苷苷 b 3 3

Therefore, tan t 苷 tan共t 兲. 72.

y

a苷1 y y 苷 x x

1 1 1 sin t 1 sin t 苷 1 sin t 1 sin t 共1 sin t兲共1 sin t兲

40. y 苷

1 x sin 2 3

2 cos2t

48. y 苷

3 t 2, tan t is negative. Thus 2 tan t 苷兹sec2 t 1.

3 cos 5x 4

period 苷

Because

兩

冉冊冉冊

36 60. Because the graph passes through the origin, we start

with an equation of the form y 苷 a sin bx. The graph 4p 4p completes one cycle in units. Thus the period is . 3 3 2p 4p Use the equation to solve for b. 苷 b 3 2p 4p 苷 b 3 • Multiply each side by 3b. 6p 苷 4bp 3 • Divide each side by 4 . 苷b 2 3 The graph has a maximum height of and a 2 3 3 minimum height of . Thus its amplitude is a 苷 . 2 2

冉冊

6.5 36 6

⬇ 41共0.9659258263兲 36 ⬇ 75.6 F

Exercise Set 2.5, page 177

period 苷 2

3π x

July 20 is represented by t 苷 6.5.

3 3 苷 2 2

y

−3

苷 15.5 F

兩兩

兩

x

π 5

3π 2

苷 41共0.5兲 36

a苷

2π 5

1

2 x 3

T共2兲苷 41 cos 2 36 6

3 sin x 2

y

2 2 苷 b 5

54. y 苷 3 sin

82. March 5 is represented by t 苷 2.

6 x

2 2 苷 b 兾3 苷 6

3 3 a苷苷 4 4

tan t 苷兹sec2 t 1

22. y 苷

x

y

兩兩

tan2 t 苷 sec2 t 1

T共6.5兲苷 41 cos

2 1 3

−1

3

period 苷

78. 1 tan2 t 苷 sec2 t

3

1 3

1 a苷 2

苷 2 sec2t

苷 41 cos

1

1

2 苷 1 sin2t 苷

3 x

1

−1

y 2 1 −1 −2

π

2π x

Substitute y苷

3 3 for a and for b in y 苷 a sin bx to produce 2 2

3 3 sin x. 2 2

S8

Solutions to the Try Exercises

冉冊

Exercise Set 2.6, page 189

40. y 苷 2 sin

1 24. y 苷 tan x 3

y

period 苷苷 b −π

34. y 苷

y

period 苷

phase shift 苷 48. y 苷 csc

1 π 4

3 c 苷 b 2

−2 −4

x

π 2

x 4 3

x

2

4

6

y

period 苷 6

8

4 y 4

2 2 苷苷4 b 兾2

3π

2 2

4

x

52. y 苷 y

2 2 苷苷 4 b 1兾2

3

冊

a苷1

−1

− 4π

y= π

2π 3 π 6

c b

phase shift 苷

c b

苷苷 1

y = cos x π

y

y = cos x

2

y = −sin x 3π 2π

−2

1 −1

π 2

π

x

2π

x

π

y

3π x 2

x 2

+ cos x

56. y 苷 cos x sin x 7π 6

兾3 苷苷 2 6 period 苷

x 2

−1

y

period 苷

22. y 苷 tan共x 兲

y=

1

1

phase shift 苷

x cos x 2

y

4π x

Exercise Set 2.7, page 199

冉

2

y

x 2

20. y 苷 cos 2x

y 4

period 苷 1

π 6

x 44. y 苷 sec 2 period 苷

1 2 苷 2兾

a苷3

π 6

2π

−4

42. y 苷 3 cos共2x 3兲 1 x

period 苷苷 b 2

40. y 苷 3 csc

苷

4

1 cot 2x 2

π

x

c phase shift 苷 b

π 2

2

−

2

period 苷 4 x

period 苷苷 b 3

y 4

a苷2

1

32. y 苷 3 tan 3x

x 1 2 2

4π

y = −sin x + cos x

x

6π x

Solutions to the Try Exercises

64. a. Phase shift 苷

period 苷

冉冊冉冊

c 苷 b

7 12

6

34.

0

7.05

− 12.5

冉冊

t . Because the phase shift 6 is 3.5 months, shift the graph of y1 3.5 units to the right to produce the graph of y2 . Now shift the graph of y2 upward 4 units to produce the graph of S.

b. First graph y1 苷 2.7 cos

a. f has pseudoperiod

2 苷 2 . 1

10 共2兲 ⬇ 1.59 Thus f completes only one full oscillation on 0 t 10.

y

Shoe sales in hundreds of pairs

12.5

苷 3.5 months,

2 2 苷苷 12 months b 兾6

b. The following graph of f shows that 兩 f 共t兲兩 0.01 for

S

6

t 10.5.

4 3.5

.012

−2

6

10

y1

y2

y = .01

t

f

8

Time in months

15.05 y = −.01

(10.5, −.01)

−.012

c. 3.5 months after January 1 is the middle of April. 78. y 苷 x cos x

Exercise Set 3.1, page 222

13

2. We will try to verify the identity by rewriting the left

side so that it involves only sines and cosines.

−4π

4π

tan x sec x sin x 苷 − 13

苷苷

Exercise Set 2.8, page 208 20. Amplitude 苷 3, frequency 苷

Because

1 , period 苷 p p

2 苷 , we have b 苷 2. Thus y 苷 3 cos 2t. b

冑

k 1 苷 m 2

冑

3 1 1 1 苷苷 , period 苷 6 27 2 3 6

冋冉冊册

1 1 y 苷 a cos 2 ft 苷 1.5 cos 2 t 苷 1.5 cos t 6 3

sin x 1 sin x cos x cos x sin2 x cos2 x

冉冊 sin x cos x

2

苷共tan x兲2 苷 tan2 x 12. sin4 x cos4 x 苷共sin2 x cos2 x兲共sin2 x cos2 x兲

苷 1共sin2 x cos2 x兲苷 sin2 x cos2 x

28. Amplitude 苷兩1.5兩苷 1.5

1 f苷 2

S9

24.

2 sin x cot x sin x 4 cot x 2 2 cot x 1 苷

共sin x兲共2 cot x 1兲 2共2 cot x 1兲 2 cot x 1

苷

共2 cot x 1兲共sin x 2兲苷 sin x 2 2 cot x 1

S10

Solutions to the Try Exercises 20. The value of a given trigonometric function of , meas-

1 1 1 1 sin x cos x sin x cos x sin x cos x 34. 苷 1 1 1 1 sin x cos x sin x cos x sin x cos x 苷

ured in degrees, is equal to its cofunction of 90 . Thus

cos 80 苷 sin共90 80 兲

cos x sin x cos x sin x

苷 sin 10

cos x sin x cos x sin x 苷 cos x sin x cos x sin x

26. sin x cos 3x cos x sin 3x 苷 sin共x 3x兲苷 sin 4x 38. tan a 苷

24 24 7 , with 0 a 90 ; sin a 苷 , cos 苷 7 25 25

苷

cos2 x sin2 x cos x 2 sin x cos x sin2 x

sin 苷

苷

cos2 x sin2 x 1 2 sin x cos x

cos 苷

2

44. Rewrite the left side so that it involves only sines and

cosines.

冉冊冉冊冉冊冉冊 24 25

15 7 17 25

8 17

360 56 416 苷 425 425 425

b. cos共兲苷 cos cos sin sin

cos x sin x 苷 cos x cos x sin x sin x sin x cos x cos x sin x

苷

冉冊冉冊冉冊冉冊 7 25

苷

cos x sin x 苷 cos2 x sin2 x sin x cos x 2

c. tan共兲苷

15 24 17 25

8 17

105 192 87 苷 425 425 425

tan tan 1 tan tan 24 8 7 15

8 24 7 15 105 苷苷 24 8 192 105 1 1 7 15 105 360 56 304 苷苷 105 192 297

冉冊冉冊

cos x sin x 苷 1 sin x cos x 2

cos x sin x cos x sin x 1

50. cos共兲苷 cos cos sin sin

苷共cos 兲共1兲共sin 兲共0兲苷 cos

苷 2 cos x 2

62. cos 5x cos 3x sin 5x sin 3x 苷 cos共5x 3x兲苷 cos 2x

Exercise Set 3.2, page 233

苷 cos共x x兲苷 cos x cos x sin x sin x

4. Use the identity cos共兲苷 cos cos sin sin

with 苷 120 and 苷 45 .

cos共120 45 兲苷 cos 120 cos 45 sin 120 sin 45

冉冊冉冊冉冊冉冊兹2 兹3 2 2

兹2 兹6 苷 4 4 苷

a. sin共兲苷 sin cos cos sin

苷

2

1 苷 2

15 8 , tan 苷 17 15

苷

cos x 2 2 cot x sin x 苷 cot x tan x cos x sin x sin x cos x

苷2

8 , with 180 270 17

兹6 兹2 4

兹2 2

苷 cos2 x sin2 x 76. sin共 2兲苷 sin cos 2 cos sin 2

苷共sin 兲共1兲共cos 兲共0兲苷 sin

Exercise Set 3.3, page 243 2. 2 sin 3 cos 3 苷 sin关2共3兲兴苷 sin 6

S11

Solutions to the Try Exercises 10. cos a 苷

24 with 270 a 360 25

冑冉冊 24 25

1

sin a 苷

2

tan a 苷

7兾25 24兾25 7 24

7 25

苷

sin 2a 苷 2 sin a cos a

cos 2a 苷 cos2 a sin2 a

苷

冉冊冉冊

苷2 苷 tan 2a 苷

24 25

7 25

336 625

1 共330°兲, we can find cos 165° by using 2 a the half-angle identity for cos with a 苷 330°. The 2 a angle 苷 165° lies in Quadrant II and the cosine func2 tion is negative in Quadrant II. Thus cos 165° 0, and we must select the minus sign that precedes the radical

28. Because 165° 苷

冑冑

苷

冉冊冉冊

in cos

苷

527 625

cos 165° 苷

2

24 25

7 25

2

a 苷

2

苷

苷苷

苷 sin x 共cos x兲

苷苷

冉冉

2

冊冉冊冉冉

冊

1 cos 2x 2

1 cos 2x 2

1 cos 2x 2

1 2 cos 2x cos2 2x 4

2

苷

冊冊

冊

2 兹3 4 2

38. Because a is in Quadrant III, cos a 0. We now solve for

cos a.

1 2 4 cos 2x 1 cos 4x 苷共1 cos 2x兲 8 2 苷

冑冉冊冑

苷

1 共 3 4 cos 2x cos 4x 3 cos 2x 4 cos2 2x 16 cos 2x cos 4x兲 1 2 苷共3 cos 2x cos 4x 4 cos 2x cos 2x cos 4x兲 16 苷

冉

7 1 25

cos a 苷兹1 sin2 a 苷

1 共1 cos 2x兲共3 4 cos 2x cos 4x兲 16

冉

冊

1 2 兹3 2 2

兹2 兹3

苷

1 1 cos 4x 共1 cos 2x兲 1 2 cos 2x 8 2

冉

2 兹3 2 2 2

冑冉冑

22. sin2 x cos4 x

苷

兹3 2 2

1

冉冊冉冊 2

1 cos 330° 2

冑冑

2 tan a 1 tan2 a 7 2 24

7 12 576 336 苷苷苷 2 49 576 527 7 1 1 576 24

2

1 cos a to produce the correct result. 2

冊

冊

Because 180° a 270°, we know that 90°

1 1 cos 4x 3 cos 2x cos 4x 4 cos 2x cos 4x 16 2

Thus

苷

1 共3 cos 2x cos 4x 2共1 cos 4x兲 cos 2x cos 4x兲 16

tan

1 苷共1 cos 2x cos 4x cos 2x cos 4x兲 16

49 625

24 苷 25

苷

1 苷共3 cos 2x cos 4x 2 2 cos 4x cos 2x cos 4x兲 16

1

2

a a a is in Quadrant II, sin 0, cos 0, and 2 2 2

a 0. 2

冑

Use the half-angle formulas.

sin

a 135°. 2

a 苷 2 苷

冑冑

1 cos a 苷 2 25 24 苷 50

冑

冉冊

24 1 25 2

49 7兹2 苷 50 10

Continued

䉴

S12

cos

Solutions to the Try Exercises

冑

冑冑

a 苷 2

苷

1 cos a 苷 2

冑

25 24 苷 50

冉冊

24 1 25 2

1 兹2 苷 50 10

7 25 24 1 25 7 25 苷苷 7 1 25

a sin a tan 苷苷 2 1 cos a

50.

冉冊

1 1 苷 1 cos 2x 1 1 2 sin2 x 苷

68. cos2

1 1 苷 csc2 x 2 sin2 x 2

冋冑

册

1 cos x 2

x 苷

2

2

1 cos x 苷 2

sec x 1 2 sec x

sin 苷

Thus 苷

cos 苷

1 2

. 3

y 苷 k sin共x 兲

冉冊 3

y 苷 2 sin x

70. From Exercise 62, we know that

冉冊

y 苷 sin x 兹3 cos x 苷 2 sin x

3

The graph of y 苷 sin x 兹3 cos x has an amplitude p of 2, and a phase shift of . It is the graph of 3 y 苷 2 sin x shifted units to the left. 3 y

−π 3

2π 3

x

5π 3

3 5 3 5 cos 2 2

2. y 苷 sin1

1 36. sin 5x cos 3x 苷关sin共5x 3x兲 sin共5x 3x兲兴 2 1 苷共sin 8x sin 2x兲 2 1 共2 sin 4x cos 4x 2 sin x cos x兲 2

Thus y 苷

sin 4x sin x 苷苷 tan x sin 4x cos x

for

y 2 2

. 4

28. Because tan共tan1 x兲苷 x for all real numbers x, we have

冋冉冊册

tan tan1

1 2

50. Let x 苷 cos1

苷 sin 4x cos 4x sin x cos x 5x 3x 5x 3x sin 2 sin cos 5x cos 3x 2 2 44. 苷 5x 3x 5x 3x sin 5x sin 3x 2 sin cos 2 2

兹2 implies 2

兹2 2

sin y 苷

苷 2 cos 4 cos共兲苷 2 cos 4 cos

苷

兹3 and 2

Exercise Set 3.5, page 265

Exercise Set 3.4, page 251 22. cos 3 cos 5 苷 2 cos

兹共兹3兲2 共1兲2 苷 2. Thus is a first-

quadrant angle.

2

1 cos x sec x 苷 2 sec x 苷

62. a 苷 1, b 苷兹3 , k 苷

cos x 苷

3 5

冉冉

56. y 苷 cos sin1

Let 苷 sin1

1 . 2

3 . Thus 5

and

y 苷 tan cos1

苷

3 5

冊

sin x 苷

冑冉冊 1

苷 tan x 苷

冊

3 5

2

苷

4 5

4 sin x 4兾5 苷苷 cos x 3兾5 3

3 5 cos1 4 13

3 3 , sin 苷 , cos 苷 4 4

冑冉冊 1

3 4

2

苷

兹7 . 4

Solutions to the Try Exercises

苷 cos1

5 5 , cos 苷 , sin 苷 13 13

冑冉冊 1

5 13

2

苷

b. Although the water rises 0.1 foot in each case, there is

12 . 13

more surface area (and thus more volume of water) at the 4.9- to 5.0-foot level near the diameter of the cylinder than at the 0.1- to 0.2-foot level near the bottom.

y 苷 cos共兲苷 cos cos sin sin

c.

3 12 5兹7 36 5兹7 36 兹7 5 苷苷 4 13 4 13 52 52 52

苷

66. sin1 x cos1

冉冊

冉

4 x 苷 sin cos cos1 6 5

p 4 cos1 6 5

5 共4兲关5 共4兲兴兹10共4兲共4兲2 5 1 5

冊冉

d.

4 cos sin cos1 6 5

兹24

1100

冊

V V = 288 0 Intersection

X=3.4476991 Y=288

10

−150

72. Let a 苷 cos1 x and 苷 cos1共x兲. Thus cos 苷 x

When V 苷 288 cubic feet, x ⬇ 3.45 feet.

and cos 苷 x. We know that sin 苷兹1 x 2 and sin 苷兹1 x 2 because is in Quadrant I and is in Quadrant II.

Exercise Set 3.6, page 278

cos1 x cos1共x兲

14.

2 cos2 x 1 苷 3 cos x 2 cos2 x 3 cos x 1 苷 0

苷苷 cos 关cos共兲兴

共2 cos x 1兲共cos x 1兲苷 0

苷 cos 共cos cos sin sin 兲

2 cos x 1 苷 0

1 1

苷 cos

关 x共x兲兹1 x

1 1

2

兹1 x

2

兴

苷 cos 共x 1 x 兲 2

2

76. The graph of y 苷 f 共x a兲 is a horizontal shift of the

graph of y 苷 f 共x兲. Therefore, the graph of y 苷 cos1共x 1兲 is the graph of y 苷 cos1 x shifted 1 unit to the right. y

y = cos−1 x

−2

84. a.

1 2

cos x 1 苷 0 cos x 苷 1

2p 4p , 3 3

x苷p

2 The solutions in the interval 0 x 2 are , , and 3 4 . 3 52. sin x 2 cos x 苷 1

sin x 苷 1 2 cos x 共sin x兲2 苷共1 2 cos x兲2 sin2 x 苷 1 4 cos x 4 cos2 x

2 x

1 cos2 x 苷 1 4 cos x 4 cos2 x 0 苷 cos x共5 cos x 4兲 cos x 苷 0

V

−150

x苷

y = cos−1 (x − 1)

1100

0

or

cos x 苷

苷 cos1共1兲苷

π

册

⬇ 352.04 cubic feet

1 4 4 3兹3 兹3 3 苷 2 5 2 5 10

苷

冉冊冉冊册

苷 12 25 cos1

1

sin共sin1 x兲苷 sin

冋冋

V共4兲苷 12 25 cos1

p 4 苷 5 6

4 p cos1 sin x 苷 6 5

S13

or

x 苷 90 , 270 10

5 cos x 4 苷 0 cos x 苷

4 5

x ⬇ 36.9 , 323.1 The solutions in the interval 0 x 360 are 90 and 323.1 . (Note: x 苷 270 and x 苷 36.9 are extraneous solutions. Neither of these values satisfies the original equation.)

S14

Solutions to the Try Exercises

56. 2 cos2 x 5 cos x 5 苷 0

The sine regression function is Y1 ⬇ 1.5347 sin共0.01624x 1.1316兲 18.4118

5 兹共5兲2 4共2兲共5兲 5 兹65 cos x 苷苷 2共2兲 4 cos x ⬇ 3.27

or

b. Using the “value” command in the CALC menu gives

Y1 共73兲 ⬇ 18.494402

cos x ⬇ 0.7656

⬇ 18:30 共to the nearest minute兲

x ⬇ 140.0 , 220.0

no solution

21

The solutions in the interval 0 x 360 are 140.0 and 220.0 . 66. cos 2x 苷

Y1=1.5346578145031*sin(.0…

兹3 2

2x 苷

5 2k 6

x苷

5 k 12

or

2x 苷

or x 苷

7 2k, k an integer 6

7 k, k an integer 12

84. 2 sin x cos x 2兹2 sin x 兹3 cos x 兹6 苷 0

1 X=73 15

Exercise Set 4.1, page 298 4.

C

2 sin x共 cos x 兹2 兲兹3 共 cos x 兹2 兲苷 0

共 cos x 兹2 兲共 2 sin x 兹3 兲苷 0

cos x 苷兹2 or

sin x 苷 x苷

no solution

B

2 , 3 3

A 苷 180 78 28 苷 74

28°

terval 关0, 2兲 are x 苷 0 and x 苷 1.895.

2π

X=1.8954943 Y=1.8954943 −3

92. When 苷 45 , d attains its maximum of 4394.5 feet. 94. a. The following work shows the sine regression proce-

dure for a TI-83/TI-83 Plus/TI-84 Plus calculator. Note that each time (given in hours:minutes) must be converted to hours. Thus 6 17:06 is 17 苷 17.1 hours. 60

44

20.

A

b c 苷 sin B sin C

a c 苷 sin A sin C

b 44 苷 sin 28 sin 78

a 44 苷 sin 74 sin 78

b苷

3

L2 17.1 17.683 18.25 18.817 19.367 19.85 20.017

b

兹3 2

86. The following graph shows that the solutions in the in-

L1 1 32 60 91 121 152 182 L1(1) = 1

78°

a

2 The solutions in the interval 0 x 2 are and . 3 3

0

Y=18.494402 365

44 sin 28 ⬇ 21 sin 78

a苷

44 sin 74 ⬇ 43 sin 78

a b 苷 sin A sin B 13.8 5.55 苷 sin A sin 22.6 sin A ⬇ 0.9555 A ⬇ 72.9 or

107.1

If A 苷 72.9 , C ⬇ 180 72.9 22.6 苷 84.5 . c 5.55 苷 sin 84.5 sin 22.6 c苷

5.55 sin 84.5 ⬇ 14.4 sin 22.6

L3

If A 苷 107.1 , C ⬇ 180 107.1 22.6 苷 50.3 .

SinReg 16, L1, L2, 35.25, Y1

c 5.55 苷 sin 50.3 sin 22.6

SinReg y = a*sin(bx+c)+d a = 1.534657815 b = .0162395434 c = -1.131646526 d = 18.41181546

c苷

5.55 sin 50.3 ⬇ 11.1 sin 22.6

Case 1: A 苷 72.9 , C 苷 84.5 , and c 苷 14.4 Case 2: A 苷 107.1 , C 苷 50.3 , and c 苷 11.1

S15

Solutions to the Try Exercises 30. The angle with its vertex at the position of the helicopter

measures 180 共59.0 77.2 兲苷 43.8 . Let the distance from the helicopter to the carrier be x. Using the Law of Sines, we have

32. A 苷 180 102 27 苷 51

x 7620 苷 sin 77.2 sin 43.8 x苷

40.

N

a2 sin B sin C 2 sin A

K苷

8.52 sin 102 sin 27 ⬇ 21 square units 2 sin 51

40. s 苷

7620 sin 77.2 sin 43.8

⬇ 10,700 feet

K苷

共to three significant digits兲

苷

苷兹19.45共19.45 10.2兲共19.45 13.3兲共19.45 15.4兲

Fire

⬇ 66.9 square units

8°

50°

1 共10.2 13.3 15.4兲苷 19.45 2

K 苷兹s共s a兲共s b兲共s c兲

C b

1 共a b c兲 2

52.

C

A

65° 20 α = 65°

32° 64 miles

B

A

α

254°

66 miles

苷 65

B

B 苷 65 8 苷 73

苷 270 254 苷 16

A 苷 180 50 65 苷 65

A 苷 16 90 32 苷 138

C 苷 180 65 73 苷 42

b 苷 4 16 苷 64 miles

b c 苷 sin B sin C b 20 苷 sin 73 sin 42 20 sin 73 b苷 sin 42 b ⬇ 29 miles

Exercise Set 4.2, page 308 12. c 2 苷 a2 b2 2ab cos C

c 2 苷 14.22 9.30 2 2共14.2兲共9.30兲 cos 9.20 c 苷兹14.22 9.30 2 2共14.2兲共9.30兲 cos 9.20 c ⬇ 5.24 18. cos A 苷

b2 c 2 a 2 2bc

1322 160 2 1082 cos A 苷 ⬇ 0.7424 2共132兲共160兲 A ⬇ cos1共0.7424兲 ⬇ 42.1 1 30. K 苷 ac sin B 2 K苷

1 共32兲共25兲 sin 127 ⬇ 320 square units 2

c 苷 3 22 苷 66 miles a2 苷 b2 c2 2bc cos A a2 苷 642 662 2共64兲共66兲 cos 138 a 苷兹642 662 2共64兲共66兲 cos 138 a ⬇ 120 miles 1 60. S 苷共324 412 516兲苷 626 2 K 苷兹626共626 324兲共626 412兲共626 516兲苷兹4,450,284,080 Cost 苷 4.15共兹4,450,284,080 兲 ⬇ $276,848, or $277,000 to the nearest $1000

Exercise Set 4.3, page 325 10. a 苷 3 3 苷 0

b 苷 0 共2兲苷 2 A vector equivalent to P 1 P 2 is v 苷具0, 2典. 12. 兩兩v兩兩苷兹62 10 2

苷兹36 100 苷兹136 苷 2兹34

苷 tan1

10 5 苷 tan1 ⬇ 59.0 6 3

Thus v has a direction of about 59 as measured from the positive x-axis. Continued

䉴

S16

Solutions to the Try Exercises

A unit vector in the direction of v is u苷 24.

冓

冔冓

冔

48. In the following figure, we need to find 储 F 1 储 and 储 F2 储 .

10 6 3兹34 5兹34 , 苷 , 34 34 2兹34 2兹34

3 3 u 2v 苷具2, 4典 2具3, 2典 4 4

冓冔冓冔

31.8° 58.2°

3 苷 , 3 具6, 4典 2 苷

9 ,7 2

苷共9i 6j兲共4i 6j兲苷共9 4兲i 共6 6兲j 苷 5i 0j 苷 5i

8 ⬇ 0.9 7 v 苷 a1 i a2 j ⬇ 1.8i 0.9j y

B

60° D x

θ A

a. sin 31.8° 苷

储 F1 储储w储储 F1 储 . 811

• 兩兩 w 兩兩苷 811

储 F1 储苷 811 sin 31.8

a2 苷 2 sin

α

From the right triangle formed by w, F1, and F2, we see 储 F2 储储 F1 储 that sin 31.8° 苷 and cos 31.8° 苷 . 储w储储w储

sin 31.8° 苷

8 ⬇ 1.8 40. a1 苷 2 cos 7

C

w F1

30. 3u 2v 苷 3共3i 2j兲 2共2i 3j兲

44.

F2

31.8°

327°

⬇ 427 Now 储 F 1 储苷储 F 1 储 . Thus the magnitude of the force needed to keep the motorcycle from rolling down the ramp is approximately 427 pounds. 储 F2 储 b. cos 31.8° 苷储w储储 F2 储 • 兩兩 w 兩兩苷 811 cos 31.8 苷 811 储 F2 储苷 811 cos 31.8° ⬇689 The magnitude of the force the motorcycle exerts against the ramp is approximately 689 pounds. 60. v w 苷共5i 3j兲共4i 2j兲

苷 5共4兲 3共2兲苷 20 6 苷 14 AB 苷 18 cos 123 i 18 sin 123 j ⬇ 9.8i 15.1j AD 苷 4 cos 30 i 4 sin 30 j ⬇ 3.5i 2j AC 苷 AB AD ⬇ 共9.8i 15.1j兲共3.5i 2j兲苷 6.3i 17.1j 兩兩AC兩兩苷兹共6.3兲2 共17.1兲2 ⬇ 18

兩兩

苷 tan1

17.1 17.1 苷 tan1 ⬇ 70 6.3 6.3

⬇ 270 70 苷 340 The speed of the boat is about 18 miles per hour at an approximate heading of 340 .

70. cos 苷

vw 兩兩v兩兩兩兩w兩兩

cos 苷

共3i 4j兲共6i 12j兲

兹32 共4兲2 兹62 共12兲2 3共6兲共4兲共12兲 cos 苷兹25 兹180 cos 苷

66

5兹180 ⬇ 10.3

72. projw v 苷

projw v 苷

⬇ 0.9839

vw 兩兩w兩兩具7, 5典具4, 1典兹共4兲 1 2

80. W 苷兩兩F兩兩兩兩s兩兩 cos

W 苷 100 25 cos 42 W ⬇ 1858 foot-pounds

2

苷

33 兹17

苷

33兹17 ⬇ 8.0 17

S17

Solutions to the Try Exercises

兩兩

Exercise Set 5.1, page 340

2 苷 tan1

8. 6 兹1 苷 6 i

z2 苷兹2 共cos 315 i sin 315 兲苷兹2 cis 315

18. 共5 3i兲共2 9i兲苷 5 3i 2 9i 苷 3 12i

z1 兹2 cis 45 苷 z2 兹2 cis 315 苷 cis共270 兲

34. 共5 2兹16兲共1 兹25兲苷关5 2共4i兲兴共1 5i兲

苷共5 8i兲共1 5i兲苷 5 25i 8i 40i 2 苷 5 25i 8i 40共1兲苷 5 25i 8i 40

苷 cos 270 i sin 270 苷 0 共i兲苷 i

Exercise Set 5.3, page 354 6. 共2 cis 330 兲4 苷 24 cis共4 330 兲

苷 45 17i 48.

苷 16 cis 1320

8i 8 i 2 3i 苷 2 3i 2 3i 2 3i 苷苷

苷 16 cis 240 苷 16共cos 240 i sin 240 兲

16 24i 2i 3i 22 32

2

冋

苷 16

16 24i 2i 3(1) 49

16 26i 3 苷 13 苷

16.

13 26i 13(1 2i) 苷苷 1 2i 13 13

冉

wk 苷 41/3 cos

兹3 苷 tan1兹3 苷 60 1

苷苷 60 , z 苷 2 cis 60

冊冉

冊

5 5 1 兹3 i sin 苷4 i 苷 2 2i兹3 3 3 2 2

56. 兹3 i 苷 2 cis共30 兲; 1 i兹3 苷 2 cis 60

共兹3 i 兲共 1 i兹3 兲苷 2 cis共30 兲 2 cis 60 苷 2 2 cis共30 60 兲苷 4 cis 30 苷 4共cos 30 i sin 30 兲

冉

苷4

1 兹3 i 2 2

冊

兹2 兹2 i 2 2

12

苷关1共cos 135 i sin 135 兲兴 12

28. 2 2i兹3 苷 4共cos 120 i sin 120 兲

兩兩

冉

苷 1 0i, or 1

兹12 共兹3兲2 苷 2

30. z 苷 4 cos

苷 8 8i兹3

苷 cos 1620 i sin 1620

Exercise Set 5.2, page 349 苷 tan1

冉

冉冊册

1 兹3 i 2 2

苷 112关cos共12 135 兲 i sin 共12 135 兲兴

1 1 1 i 1 i 60. 83 苷 80 3 苷 3 苷 3 苷 4 苷 i i i i i i i i

12. r 苷

1 苷 45 ; 2 苷 315 1

冊

冉冉冉

冊

r1 苷兹12 12 苷兹2

兩兩

1 苷 tan1

1 苷 45 ; 1 苷 45 1

z1 苷兹2 共cos 45 i sin 45 兲苷兹2 cis 45

120 120 i sin ⬇ 1.216 1.020i 3 3

w1 苷 41/3 cos

120 360 120 360 i sin 3 3

r2 苷兹1 共1兲苷兹2 2

冊

⬇ 1.492 0.543i w2 苷 41/3 cos

冊

120 360 2 120 360 2 i sin 3 3

⬇ 0.276 1.563i 30. 1 i兹3 苷 2共cos 120 i sin 120 兲

冉

wk 苷 21/2 cos

冊

120 360 k 120 360 k i sin , 2 2 k 苷 0, 1

w0 苷 21/2共cos 60 i sin 60 兲苷

冉

w1 苷 21/2 cos

兹2 兹6 i 2 2 共k 苷 0兲

共k 苷 1兲苷 2 共cos 240 i sin 240 兲 1/2

苷

冊

120 360 120 360 i sin 2 2

z2 苷 1 i 2

k 苷 0, 1, 2

w0 苷 41/3 cos

苷 2兹3 2i 60. z1 苷 1 i

冊

120 360 k 120 360 k i sin , 3 3

兹2 兹6 i 2 2

S18

Solutions to the Try Exercises Center: 共0, 0兲

Exercise Set 6.1, page 370 6. Comparing x 2 苷 4py with x 2 苷

1 . 16

or p 苷

1 1 y, we have 4p 苷 , 4 4

冉冊

x

28.

9共x 2 4x兲 16共 y 2 y兲苷 104

冉

x 5x 苷 4y 1 25 25 x 5x 苷 4y 1 4 4 2

冉冊冉 2

苷4 y

29 16

冊

4p 苷 4

• Complete the square. •h

冊冉

5 29 , 2 16

冊

5 13 Focus: 共h, k p兲苷 , 2 16

共x 2兲 16 2

2

苷 104 36 4 苷 144

2

苷1

1 2

y

4

a 苷 4, b 苷 3, y

c 苷兹42 32 苷兹7

2

Vertices: 2,

45 16

2

x

30. Vertex: 共0, 0兲, focus: 共5, 0兲; p 苷 5 because focus is 共 p, 0兲.

y 2 苷 4px

冉冊冉冊冉冊冉 1 2

and 6,

Foci: 2 兹7,

1 2

2

1 2

and 2 兹7,

1 2

x

冊

44. Center 共4, 1兲苷共h, k兲. Therefore, h 苷 4 and k 苷 1.

Length of minor axis is 8, so 2b 苷 8, or b 苷 4. The equation of the ellipse is of the form

y 2 苷 4共5兲x y 2 苷 20x 32. Vertex: 共2, 3兲, focus: 共0, 3兲

共h, k兲苷共2, 3兲, so h 苷 2 and k 苷 3. Focus is 共h p, k兲苷共2 p, 3兲苷共0, 3兲. Therefore, 2 p 苷 0 and p 苷 2. 共 y k兲苷 4p共x h兲 2

共x h兲2 共 y k兲2 苷1 a2 b2 共x 4兲2 共 y 1兲2 苷1 a2 16 共0 4兲2 共4 1兲2 苷1 a2 16

共 y 3兲2 苷 4共2兲共x 2兲共 y 3兲2 苷 8共x 2兲 x 苷 4py 2

2

40.5 64

a2 苷

p ⬇ 25.6 feet

Exercise Set 6.2, page 385 22. 25x 2 12y 2 苷 300 • a2 25, b2 12, c2 25 12

y2 x2 苷1 12 25

16 9 苷1 a2 16

• h 4, k 1, b 4 • The point (0, 4) is on the graph. Thus x 0 and y 4 satisfy the equation. • Solve for a 2.

16 7 苷 a2 16

40.52 苷 4p共16兲 p苷

9

冉冊

Center: 2,

1 4

1 2 1 y 2

9共x 2兲2 16 y

5 29 ,k 2 16

• Compare to 共x h兲2 4p共 y k兲2.

p苷1

Directrix: y 苷 k p 苷

冊冉冊冉冊

9共x 2 4x 4兲 16 y 2 y

2

5 2

9x 2 16y 2 36x 16y 104 苷 0 9x 2 36x 16y 2 16y 104 苷 0

−4

22. x 2 5x 4y 1 苷 0

40.

x

1

1 16

Directrix: y 苷

冉

2

y

1 Focus: 0, 16

Vertex:

Foci: 共 0, 兹13 兲 and 共 0, 兹13 兲

2

Vertex: 共0, 0兲

x

y

Vertices: 共0, 5兲 and 共0, 5兲

a 5, b 2兹3, c 兹13

256 7

共x 4兲2 共 y 1兲2 苷1 256兾7 16 50. Because the foci are 共0, 3兲 and 共0, 3兲, c 苷 3 and the cen-

ter is 共0, 0兲, the midpoint of the line segment between 共0, 3兲 and 共0, 3兲.

S19

Solutions to the Try Exercises c a

e苷

Vertices: 共h, k a兲苷共1, 1兲共h, k a兲苷共1, 7兲

1 3 苷 4 a

1 •e 4

c 苷 a b2 苷 16 9 苷 25 2

c 苷兹25 苷 5

a 苷 12 32 苷 122 b2

• c 2 a2 b2

b2 苷 144 9 苷 135

• Solve for b 2.

Foci: 共h, k c兲苷共1, 2兲共h, k c兲苷共1, 8兲 Because b2 苷 9 and b 苷 3, the asymptotes are

x2 y2 The equation of the ellipse is 苷 1. 135 144 Aphelion 苷 a c 苷 67.58 million miles Thus c 苷 67.58 a 苷 0.50 million miles. An equation of the orbit of Venus is x2 y2 苷 1. 2 67.08 67.0782 60. The length of the semimajor axis is 50 feet. Thus 2

b 苷兹50 2 322

and c 苷 150.

Exercise Set 6.3, page 399

2a 苷 rate time 2a 苷 0.186 800 苷 148.8 miles

y2 x2 苷1 25 36

Thus a 苷 74.4 miles.

a2 苷 25

b 苷兹c2 a2

b2 苷 36 c2 苷 a2 b2 苷 25 36 苷 61 b苷6

苷兹150 2 74.42 ⬇ 130.25 miles

c 苷兹61

Transverse axis is on y-axis because y 2 term is positive.

The ship is located on the hyperbola given by

Center: 共0, 0兲

x2 y2 苷1 2 74.4 130.252

Foci: 共 0, 兹61 兲 and 共 0, 兹61 兲 Asymptotes: y 苷 y苷

y

5 x and 6

b. The ship will reach the coastline when x 0 and

y 苷 0. Thus

4 8 x

5 x 6

16x 2 9y 2 32x 54y 79 苷 0 16共x 2x 1兲 9共 y 6y 9兲苷 79 16 81 2

x2 02 苷1 74.42 130.252 x2 苷1 74.42

Vertices: 共0, 5兲 and 共0, 5兲 28.

center is 共0, 3兲. c e苷 c2 苷 a2 b2 a 52 苷 22 b 2 5 c b2 苷 25 4 苷 21 苷 2 2

56. a. Because the transmitters are 300 miles apart, 2c 苷 300

b ⬇ 38.4 feet

a苷5

x

x 2 共 y 3兲2 苷 1. 4 21

b2 苷 50 2 322

6.

1

c苷5 Substituting into the standard equation yields

c 2 苷 a 2 b2 32 苷 50 b

2

50. Because the vertices are 共2, 3兲 and 共2, 3兲, a 苷 2 and the

b 苷兹a2 c2 苷兹67.082 0.50 2 ⬇ 67.078

2

y

4 y 3 苷共x 1兲 and 3 4 y 3 苷共x 1兲. 3

58. The mean distance is a 苷 67.08 million miles.

2

2

2

苷 144 共 y 3兲2 共x 1兲2 苷1 16 9 Transverse axis is parallel to y-axis because y 2 term is positive. Center is at 共1, 3兲; a2 苷 16 so a 苷 4.

x 2 苷 74.42 x 苷 74.4 The ship reaches the coastline 74.4 miles to the left of the origin, at the point 共74.4, 0兲.

S20

Solutions to the Try Exercises

Exercise Set 6.4, page 410 xy 苷 10

sin 苷

xy 10 苷 0 A 苷 0, B 苷 1, C 苷 0, F 苷 10 cot 2 苷

AC 00 苷苷0 B 1

cos 苷

冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊 1

兹2 2

兹2 2

0

兹2 2

2

苷

1 2

兹2 2

2

1

兹2 2

兹2 2

0

兹2 2

2

苷

1 2

共 x 兲 1 1 共 y 兲共 x 兲2 共 y 兲2 10 苷 0 or 苷1 2 2 20 20

苷 2兹5

兹5 5

4

冊冉冊冉冉冉

冊冉冊冊冉冊冊冊

2兹5 5

2

兹5 5

4

4

2兹5 5

4

兹5 5

兹5

2兹5 5

苷0

E 苷 D sin E cos

8

45°

4

2

2兹5 5

苷 2兹5

2兹5 5

2

苷5

2兹5 5

兹5

兹5 5

兹5 5

2

苷0

苷5

5共x 兲2 5y 苷 0 or y 苷共x 兲2

x 4

y 8

−8

− 8

兹5 5

D 苷 D cos E sin

x'

y' 8

兹5 5

苷1

2

y

63.4°

y'

−

苷

C 苷 A sin2 B cos sin C cos2

F 苷 F 苷 10 2

冉冊冉冉冊冉冉冊冉冊

苷1

C 苷 A sin2 B cos sin C cos2 苷0

2

2兹5 5

A 苷 A cos2 B cos sin C sin2

A 苷 A cos B cos sin C sin 2

2

3 5

苷

⬇ 63.4

2

兹2 2

2

1

Thus 2 苷 90 and 苷 45 .

苷0

冑冑

3 5

x'

10.

冉冊冉冊

1

8 x

−8

20. x 4xy 4y 2兹5x 兹5y 苷 0 2

2

−8

A 苷 1, B 苷 4, C 苷 4, D 苷 2兹5, E 苷兹5, F 苷 0 cot 2 苷

AC 14 3 苷苷 B 4 4

The following figure shows the second quadrant angle 3 2 for which cot 2 苷 . 4

26. Use y1 and y2 as in Equations (12) and (13) that precede

Example 4. Store the following constants. A 苷 2, B 苷 8, C 苷 8, D 苷 20, E 苷 24, F 苷 3 Graph y1 and y2 on the same screen to produce the following parabola.

y

5

(−3, 4)

4

5

2 −5

4 −1

−3

x

3 From the figure we see that cos 2 苷 . 5 Use the half-angle formulas to find sin a and cos .

32. Because

B2 4AC 苷共 10兹3 兲 4共11兲共1兲 2

苷 300 44 0 the graph is a hyperbola.

Solutions to the Try Exercises

Exercise Set 6.5, page 423 14.

兩兩

苷 tan1

48. r 苷兹x 2 y 2

苷兹共12兲2 共5兲2 苷兹144 25 苷 13

5 ⬇ 22.6 12

is a Quadrant IV angle. Thus ⬇ 360 22.6 苷 337.4 .

16. r 苷 5 cos 3

The approximate polar coordinates of the point are 共13, 337.4 兲.

Because 3 is odd, this is a rose with three petals. 50. 8

20. Because 兩a兩苷兩b兩苷 2, the graph of r 苷 2 2 cos ,

, is a cardioid.

r 苷 2 sin r 2 苷 2r sin x 2 y 2 苷 2y 2 x (y 2 2y) 苷 0 2 x (y 2 2y 1) 苷 1 x 2 (y 1)2 苷 12

• Multiply each side by r. • r2 x 2 y 2; y r sin • Subtract 2y from each side. • Complete the square.

A rectangular form of r 苷 2 sin is x 2 (y 1)2 苷 12. The graph of each of these equations is a circle with center (0, 1) and radius 1.

4

r 苷 cot

58.

r苷

26. r 苷 4 4 sin

min 苷 0 max 苷 2 step 苷 0.1

• is the reference angle for .

兩兩

苷 tan1

苷兹169

4

y x

S21

cos sin

• cot

cos sin

r sin 苷 cos

3.25

r共r sin 兲苷 r cos

−9.4

共兹x 2 y 2 兲y 苷 x

9.4

• Multiply each side by r. • y r sin ; x r cos

共x 2 y 2兲y 2 苷 x 2

• Square each side.

y x 2y 2 x 2 苷 0 4

−9.75

2x 3y 苷 6

70.

2r cos 3r sin 苷 6

32. r 苷 4 sec

min 苷 0 max 苷 2 step 苷 0.1

• x r cos ; y r sin

r共2 cos 3 sin 兲苷 6

6.25

r苷 −9.4

6 2 cos 3 sin

9.4

Exercise Set 6.6, page 431 −6.25

44. x 苷 r cos

冋冉冊册冉冊

苷共2兲 cos 3 苷共2兲

1 2

苷1

8 2 4 cos 4 r苷 1 2 cos

2. r 苷

y 苷 r sin

冋冉冊册冉冊

苷共2兲 sin 3 苷共2兲

兹3 2

苷兹3

The rectangular coordinates of the point are 共 1, 兹3 兲.

• Divide numerator and denominator by 2.

e 苷 2, so the graph is a hyperbola. The transverse axis is on the polar axis because the equation involves cos . Let 苷 0. r苷

8 8 苷苷 4 2 4 cos 0 2 4

S22

Solutions to the Try Exercises

Let 苷 . r苷

24. e 苷 1, r cos 苷 2

Thus the directrix is x 苷 2. The distance from the focus (pole) to the directrix is d 苷 2.

8 8 4 苷苷 2 4 cos 2 4 3

The vertices are 共4, 0兲 and

冉冊

4 , . 3

8

r苷

ed 1 e cos

r苷

共1兲共2兲 1 共1兲 cos

r苷

2 1 cos

• Use Eq (2) because the vertical directrix is to the left of the pole. • e 1, d 2

Exercise Set 6.7, page 440 6 4. r 苷 3 2 cos

6. Plotting points for several values of t yields the follow-

ing graph.

2

r苷 1

冉冊 2 3

cos

y 4

• Divide numerator and denominator by 3.

2 , so the graph is an ellipse. The major axis is on 3 the polar axis because the equation involves cos . Let 苷 0.

e苷

6 6 6 r苷苷苷 3 2 cos 0 3 2 5

r苷

6 6 苷苷6 3 2 cos 3 2

. 2

12. x 苷 3 2 cos t, y 苷 1 3 sin t, 0 t 2

x3 y1 , sin t 苷 2 3

y 4

冉冊冉 x3 2

冉冊

6 , 0 and 共6, 兲. Let 5

2

冊

y1 3

2

−4

苷1

冉冊

3 2 cos 2

苷

6

6 r苷苷苷2 30 3 3 2 cos 2

冉冊

y 苷 2 共x 1兲

(1, 2) 5x

−5

y 苷 x 3

−5

Because x 苷 1 t 2 and t 2 0 for all real numbers t, x 1 for all t. Similarly, y 2 for all t.

冉冊

Two additional points on the ellipse are 2, 2 3 2, . 2

冉冊

y 5

t2 苷 x 1

3 . 2

x

−4

x 苷 1 t2

6 苷2 30

4

共x 3兲2 共 y 1兲2 苷1 4 9 14. x 苷 1 t 2, y 苷 2 t 2, t R

6

Let 苷

4 x (−1, 1)

cos2 t sin2 t 苷 1

The vertices on the major axis are

苷

−4

cos t 苷

Let 苷 . r苷

−4

24. Eliminate the parameter to determine an equation in x

and

and y that is equivalent to the two parametric equations x 苷 t 1, y 苷兹t. First, solve x 苷 t 1 for t. x苷t1 t苷x1 Substitute x 1 for t in the equation y 苷兹t. y 苷兹x 1

8

Squaring both sides produces y 2 苷 x 1, with y 0.

S23

Solutions to the Try Exercises (Note: y is nonnegative because of the information provided by the parametric equation y 苷兹t.) Thus the graph of the parametric equations x 苷 t 1, y 苷兹t, with t 0, is the upper half of a parabola. When t 苷 0, P is at (1, 0); when t 苷 4, P is at (5, 2). Thus P starts at (1, 0) and moves along the upper portion of the parabola y 2 苷 x 1 until it reaches (5, 2) at time t 苷 4.

the points. The graph should be asymptotic to the negative portion of the x-axis, as shown in the following figure. y 6

4 2

y

−4 −2

2

4x

(5, 2) 2

28. Because F共x兲苷 6x5 苷 f 共x 5兲, the graph of F共x兲 can be (1, 0)

4

produced by shifting the graph of f horizontally to the left 5 units.

x

32. The maximum height of the cycloid is 2a 苷 2共3兲苷 6.

The cycloid intersects the x-axis every 2 a 苷 2 共3兲苷 6 units. 7

冋冉冊册

5 x 苷 f 共x兲, the graph of F共x兲 2 can be produced by reflecting the graph of f across the x-axis.

30. Because F共x兲苷

48. a. A共45兲苷 200e0.014共45兲

⬇ 106.52 −1

After 45 minutes the patient will have about 107 milligrams of medication in his or her bloodstream.

36

−1

36.

b. Use a graphing calculator to graph y 苷 200e0.014x

200

and y 苷 50 in the same viewing window, as shown below. 250

0

1200

0

The maximum height is about 195 feet when t ⬇ 3.50 seconds. The range is 1117 feet when t ⬇ 6.99 seconds.

Exercise Set 7.1, page 461 1 1 1 苷 2 苷 5 5 5 25

22. The graph of f 共x兲苷

冉冊 5 2

x

has a y-intercept of 共0, 1兲

冉冊冉冊冉冊

5 . Plot a few addi2 25 and 2, . Because the 4

and the graph passes through 1, tional points, such as 1,

2 5

5 is greater than 1, we know that the graph must 2 have all the properties of an increasing exponential function. Draw a smooth increasing curve through base

0

Intersection X=99.021026 Y=50

150

− 100

The x-coordinate (which represents time in minutes) of the point of intersection is about 99.02. Thus it will take about 99 minutes before the patient’s medication level is reduced to 50 milligrams.

2. f 共3兲苷 53 苷 5 5 5 苷 125

f 共2兲苷 52 苷

Plot1 Plot2 Plot3 \Y 1 = 2 0 0 * e ^ ( - . 0 1 4 * X ) \Y 2 = 5 0 \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 =

54. a. P共0兲苷

3600 1 7e0.05共0兲

苷

3600 17

苷

3600 8

苷 450 Immediately after the lake was stocked, the lake contained 450 bass. Continued

䉴

S24

Solutions to the Try Exercises

b. P共12兲苷

3600 1 7e0.05共12兲

3 months the student’s average typing speed was about 45 words per minute.

⬇ 743.54

b. Use the intersection feature of a graphing utility to

find the x-coordinate of the point of intersection of the graphs of y = 5 + 29 ln(x + 1) and y = 65.

After 1 year (12 months) there were about 744 bass in the lake. 7 approaches 0. Thus, as t l , e0.05t 3600 3600 will approach P共t兲苷苷 3600. As 0.05t 1 7e 10 time goes by the bass population will increase, approaching 3600.

c. As t l

, 7e0.05t 苷

75

Intersection X=6.9166293 Y=65

0

16

0

Exercise Set 7.2, page 476 The graphs intersect at about 共6.9, 65兲. The student will achieve a typing speed of 65 words per minute in about 6.9 months.

4. The exponential form of y 苷 logb x is by 苷 x. Thus the ex-

ponential form of 3 苷 log4 64 is 4 苷 64. 3

14. The logarithmic form of by 苷 x is y 苷 logb x. Thus the

logarithmic form of 53 苷 125 is 3 苷 log5 125.

32. log 1,000,000 苷 log10 10 苷 6

Exercise Set 7.3, page 489

6

44. To graph y 苷 log6 x, use the equivalent exponential

equation x 苷 6y. Choose some y values, such as 1, 0, 1, and calculate the corresponding x values. This yields the 1 ordered pairs , 1 , 共1, 0兲, and 共6, 1兲. Plot these 6 ordered pairs and draw a smooth curve through the points to produce the following graph.

冉冊 y 6

2. ln

z3 兹xy

苷 ln z 3 ln 兹xy 苷 ln z 3 ln(xy)1/2 苷 3 ln z

1 ln(xy) 2

苷 3 ln z

1 (ln x ln y) 2

苷 3 ln z

1 1 ln x ln y 2 2

18. 3 log2 t

3

3

6

1 log2 u 4 log2 v 苷 log2 t 3 log2 u1/3 log2 v 4 3

9 x

52. log4共5 x兲 is defined only for 5 x 0, which is

equivalent to x 5. Using interval notation, the domain of k共x兲苷 log4共5 x兲 is 共, 5兲.

66. The graph of f 共x兲苷 log6共x 3兲 can be produced by shift-

ing the graph of f 共x兲苷 log6 x (from Exercise 44) 3 units to the left. See the following figure.

34. log5 37 苷

苷 log2

t3 log2 v 4 u1/3

苷 log2

t 3v 4 u1/3

苷 log2

t 3v 4 3 兹u

log 37 ⬇ 2.2436 log 5

ln(5 x) ln(5 x) , so enter into Y1 on a ln 8 ln 8 graphing calculator.

46. log8(5 x) 苷

y 6

3 5

3

6 x −3

7

86. a. S(0) = 5 + 29 ln(0 + 1) = 5 + 0 = 5. When starting, the

student had an average typing speed of 5 words per minute. S共3兲苷 5 29 ln共3 1兲 ⬇ 45.2. After

−5

Solutions to the Try Exercises 70. pH 苷 log 关H 兴苷 log共1.26 10 3 兲 ⬇ 2.9 72.

x log 3 2 log 3 2x log 4 苷 log 4 x log 3 2x log 4 苷 log 4 2 log 3

pH 苷 log 关H 兴

x共log 3 2 log 4兲苷 log 4 2 log 3

5.6 苷 log 关H 兴

x苷

5.6 苷 log 关H 兴 10 5.6 苷 H The hydronium-ion concentration is 10 5.6 ⬇ 2.51 10 6 mole per liter.

冉冊冉

冊

I 398,107,000I0 苷 log 苷 log 398,107,000 I0 I0

78. M 苷 log

x ⬇ 2.141 22. log共x 19兲苷 2

x 2 19 苷 10 2 x 2 19 苷 100 x 2 苷 81 x 苷 9

冉冊

A check shows that 9 and 9 are both solutions of the original equation.

I 苷 9.5 I0

26. log3 x log3 共x 6兲苷 3

I 苷 10 9.5 I0

log3 关x 共x 6兲兴苷 3 33 苷 x共x 6兲

I 苷 10 I0 9.5

27 苷 x 2 6x

I ⬇ 3,162,277,660I0 82. In Example 7 we noticed that if an earthquake has a

Richter scale magnitude of M1 and a smaller earthquake has a Richter scale magnitude of M2 , then the larger earthquake is 10 M –M times as intense as the smaller earthquake. In this exercise, M1 苷 9.5 and M2 苷 8.3. Thus 10 M –M 苷 10 9.58.3 苷 10 1.2 ⬇ 15.8. The 1960 earthquake in Chile was about 15.8 times as intense as the San Francisco earthquake of 1906. 1

1

2

苷 log 26 3 log关8 17兴 2.92

• Substitute 26 for

⬇ 1.4150 6.4006 2.92

A and 17 for t.

⬇ 4.9

Exercise Set 7.4, page 501

Because log3 x is defined only for x 0, the only solution is x 苷 3. 36. ln x 苷

冉冊冋冉冊册冋冉冊册

1 5 ln 2x 2 2

ln x 苷

1 2

ln x 苷

5 1 ln 2 2x 2 2

ln x 苷

1 ln共4x 5兲 2

ln 2x

5 2

1 ln 2 2

ln 2

ln x 苷 ln共4x 5兲1/2 x 2 苷 4x 5

3x 苷 35

0 苷 x 2 4x 5

x苷5

0 苷共x 5兲共x 1兲

6x 苷 50

x 苷 5 or x 苷 1

log共6x 兲苷 log 50

Check:

x log 6 苷 log 50

18.

共x 9兲共x 3兲苷 0 x 苷 9 or x 苷 3

x 苷兹4x 5

2. 3x 苷 243

x 苷

x 2 6x 27 苷 0

2

86. M 苷 log A 3 log 8t 2.92

10.

log 4 2 log 3 log 3 2 log 4

2

⬇ 8.6 80. log

S25

log 50 ⬇ 2.18 log 6 3x2 苷 42x1 log 3x2 苷 log 42x1 共x 2兲log 3 苷共2x 1兲log 4 x log 3 2 log 3 苷 2x log 4 log 4

ln 5 苷

冉

1 5 ln 10 2 2

冊

1 ln 2 2

1.6094 ⬇ 1.2629 0.3466 Because ln共1兲 is not defined, 1 is not a solution. Thus the only solution is x 苷 5.

S26

Solutions to the Try Exercises 10 x 10 x 苷8 2

42.

The growth function is N(t) ⬇ 276,400e 0.013555t. The year 2008 is represented by t 苷 8.

10 x 10 x 苷 16

N(8) ⬇ 276,400 e0.0135558

x

10 共10 10 兲苷共16兲10 x

x

x

• Multiply each side by 10x.

10 2x 1 苷 16共10 x 兲 10 2x 16共10 x 兲 1 苷 0 u2 16u 1 苷 0 u苷

⬇ 308,100 The exponential growth function yields 308,100 as the approximate population of Aurora in 2008.

• Let u 10x.

16 兹16 4共1兲共1兲苷 8 3兹7 2 2

10 苷 8 3兹7 x

• Replace u with

8.

N共t兲苷 N0 e kt N共138兲苷 N0 e 138k 0.5N0 苷 N0 e 138k

10x.

log 10 苷 log共8 3兹7兲 x

0.5 苷 e138k

x 苷 log共8 3兹7兲 ⬇ 1.20241 74. a.

t苷

9 24 v ln 24 24 v

1.5 苷

9 24 v ln 24 24 v

k苷

24 v 24 v

12.

N共t兲苷 N0共0.5兲t/5730 0.65N0 苷 N0共0.5兲t/5730

• N = ln M means eN = M.

0.65 苷共0.5兲t/5730 ln 0.65 苷 ln共0.5兲t/5730

共24 v兲e 苷 24 v 4

v ve 苷 24 24e 4

t 苷 5730

4

v共1 e4 兲苷 24 24e4 v苷

The bone is approximately 3600 years old.

24 24e ⬇ 23.14 1 e4

b. The vertical asymptote is v 苷 24. c. Due to the air resistance, the object can never reach or

exceed a velocity of 24 feet per second.

Exercise Set 7.5, page 516 the year 2004. Start by substituting 4 for t in N(t) 苷 N0 e kt. N(4) 苷 N0 e k4 291,800 苷 276,400e4k

• Substitute.

291,800 苷 e4k 276,400

• Solve for k.

冊冊

291,800 1 ln 苷k 4 276,400 0.013555 ⬇ k

16. a. P 苷 12,500, r 苷 0.08, t 苷 10, n 苷 1

冉

A 苷 12,500 1 b. n 苷 365

冉冉

A 苷 12,500 1 c. n 苷 8760

6. Let t 苷 0 represent the year 2000 and let t 苷 4 represent

冉冉

ln 0.65 ⬇ 3600 ln 0.5

4

The velocity is about 23.14 feet per second.

291,800 ln 苷 4k 276,400

ln 0.5 ⬇ 0.005023 138

N共t兲苷 N0共0.5兲t/138 ⬇ N0 e 0.005023t

24 v 4 苷 ln 24 v e4 苷

ln 0.5 苷 138k

A 苷 12,500 1

冊

0.08 1

10

冊冊

0.08 365

0.08 8760

⬇ $26,986.56

3650

⬇ $27,816.82

87600

⬇ $27,819.16

18. P 苷 32,000, r 苷 0.08, t 苷 3

A 苷 Pert 苷 32,000e3共0.08兲 ⬇ $40,679.97 22. t 苷

t苷

ln 3 , r 苷 0.055 r ln 3 0.055

t ⬇ 20 years (to the nearest year)

38. a. Represent the year 2004 by t 苷 0; then the year 2005

will be represented by t 苷 1. Use the following substitutions: P0 苷 240, P共1兲苷 310, c 苷 3400, and a苷

c P0 3400 240 苷 ⬇ 13.16667. P0 240

S27

Solutions to the Try Exercises P共t兲苷

c 1 aebt

P共1兲苷

3400 1 13.16667eb共1兲

310 苷

3400 1 13.16667eb

310共1 13.16667eb 兲苷 3400 3400 1 13.16667eb 苷 310 13.16667eb 苷

3400 1 310

13.16667eb ⬇ 9.96774 eb ⬇

9.96774 13.16667

冉冊

t 苷 2 ln 1

50 ⬇ 3.0 64

The velocity is 50 feet per second in approximately 3.0 seconds.

, et/2 l 0. Therefore, 64共1 et/2 兲 l 64. The horizontal asymptote is v 苷 64.

c. As t l

d. Because of air resistance, the velocity of the object will

approach, but never reach or exceed, 64 feet per second.

Exercise Set 7.6, page 529 4. The following scatter plot suggests that the data can be

modeled by an increasing function that is concave down. Thus the most suitable model for the data is an increasing logarithmic function. 8

b ⬇ ln

9.96774 13.16667

b ⬇ 0.27833 Using a 苷 13.16667, b 苷 0.27833, and c 苷 3400 gives the following logistic model. P共t兲 ⬇

3400 1 13.16667e0.27833t

b. Because 2011 is 7 years past 2004, the year 2011 is

represented by t 苷 7. 3400 P共7兲 ⬇ ⬇ 1182 1 13.16667e0.27833共7兲

0

30

0 Xscl = 5

Yscl = 1

22. From the scatter plot in the following figure, it appears

that the data can be closely modeled by a decreasing exponential function of the form y 苷 abx, with b 1. 12

According to the model there will be about 1182 groundhogs in 2011. 48. a.

v 0

50

25

0 Xscl = 5

5

t

b. Here is an algebraic solution. An approximate

solution can be obtained from the graph. v 苷 64共1 et/2 兲

Yscl = 1

The calculator display in the following figure shows that the exponential regression equation is y ⬇10.1468(0.89104)x, where x is the altitude in kilometers and y is the pressure in newtons per square centimeter. 12

ExpReg y=a*b^x a=10.14681746 b=.8910371309 r2=.9997309204 r=-.9998654511

50 苷 64共1 et/2 兲 50 苷共1 et/2 兲 64 50 苷 et/2 1 64

冉冊

ln 1

50 t 苷 64 2

0

25

0 Xscl = 5

Yscl = 1

The correlation coefficient r ⬇ 0.99987 is close to 1. This indicates that the function y ⬇ 10.1468(0.89104)x provides a good fit for the data. The graph of y also Continued

䉴

S28

Solutions to the Try Exercises

indicates that the regression function provides a good model for the data. When x 苷 24 kilometers, the atmospheric pressure is about 10.1468(0.89104)24 ⬇ 0.6 newton per square centimeter.

26. a. Use a graphing utility to perform a logistic regression

on the data. The following figure shows the results obtained by using a TI-83/TI-83 Plus/TI-84 Plus graphing calculator.

24. a. Use a graphing utility to perform an exponential

Logistic y=c/(1+ae^(-bx)) a=2.233147667 b=.043679087 c=1534426.531

regression and a logarithmic regression. For the given data, the logarithmic function y 苷 61.735786 4.1044761 ln x provides a slightly better fit than does the exponential regression function, as determined by comparing the correlation coefficients. See the calculator displays below.

The logistic regression function for the data is 1,534,427 P(t) ⬇ . 1 2.233148 e0.043679t

50

ExpReg y=a*b^x a=48.55569695 b=.9987196271 r2=.8923900275 r=-.944663976

Y1=61.736–4.1045ln(X)

LnReg y=a+blnx a=61.73578555 b=-4.104476112 r2=.9302117833 r=-.9644748744

b. The year 2010 is represented by t 苷 60.

P共60兲苷 48

X=108

Y=42.518192

40 Xscl = 10

b. To predict the world record time in 2008, evaluate

y 苷 61.735786 4.104476 ln x at x 苷 108. The graph on the right above shows that the predicted world record time in the men’s 400-meter race for the year 2008 is about 42.52 seconds.

Yscl = 1

110

1,534,427 ⬇ 1,320,000 1 2.233148e0.043679(60)

The logistic regression function predicts that Hawaii’s population will be about 1,320,000 in 2010. c. The carrying capacity, to the nearest thousand, of the

logistic model is 1,534,000 people.

Answers to Selected Exercises Exercise Set 1.1, page 12 x 5 c 4 21. x 苷 y 5 23. y 苷 2 2 ab x 1 兹5 2 兹2 5 兹61 3 兹41 3 兹29 兹2 y苷 , 兹2 39. 27. 3, 5 29. 31. 33. 35. 37. 1x 2 2 6 4 2 2 3 7 13 3, 5 43. , 24 45. 0, 47. 2, 8 49. 共, 4兲 51. 共, 6兲 53. 共, 3兴 55. , 57. 共, 2兲 8 3 8 1 4 共, 7兲共0, 兲 61. 共5, 2兲 63. 共, 4兴关7, 兲 65. , 67. 共4, 4兲 69. 共8, 10兲 71. 共, 33兲共27, 兲 2 3 3 4 5 28 , , , 75. 共, 8兴关2, 兲 77. , 8 79. 共, 4兴 81. 共, 兲 83. 4 2 2 3 5 3.5 centimeters by 10 centimeters 87. 100 feet by 150 feet 89. at least 58 checks 91. at least 34 sales 93. 20 C 40 22 97. 共0, 210兲 99. 1 second t 3 seconds 101. a. 兩s 4.25兩 0.01 b. 4.24 s 4.26

1. 15 25. 41. 59. 73. 85. 95.

3. 4 5.

冉

9 2

7.

2 9

9. 12

11. 16

13. 75

15.

冊冉冊

1 2

17. 1200

冋册

19. y 苷

冋

冋

册

冊

冋冊

Prepare for This Section (1.2), page 15 PS1.

3 2

PS2. 5兹2

PS3. No

PS4. 16

PS5. 2

PS6. 5

Exercise Set 1.2, page 27 3. a. $31,500 b. $34,100 c. $34,111 5. 7兹5 7. 兹1261 9. 兹89 11. 兹38 12兹6 y 15. x兹10 17. 共12, 0兲, 共4, 0兲 19. 共3, 2兲 21. 共6, 4兲 23. 共0.875, 3.91兲 25.

y

1.

(2, 4)

(−2, 1)

4

x

(−5, −3)

(0, −3)

y 3

29.

4 −3 −4

4

x

−4

y

27.

13. 2兹a2 b 2

y

31.

33. 3

5

x

−3

x

y

y

35. 3

3 x

x

−2 3

x

−4

−6

37.

冉冊

39. 共6, 0兲, 0,

y

2

12 5

41. 共5, 0兲; 共 0, 兹5 兲, 共 0, 兹5 兲

y 6

y 4

x

−4

−6

6

x

−4

5

x

−4

A1

A2

Answers to Selected Exercises

43. 共4, 0兲; 共0, 4兲, 共0, 4兲

45. 共 2, 0兲, 共0, 2兲

y

y

4

4

4 x

−4

−4

4

−4

47. 共 4, 0兲, 共0, 4兲

−4

49. center 共0, 0兲, radius 6 51. center 共1, 3兲, radius 7 53. center 共2, 5兲, radius 5 1 55. center 共8, 0兲, radius 57. 共x 4兲2 共 y 1兲2 苷 22 2 1 2 1 2 2 59. x 61. 共x 0兲2 共 y 0兲2 苷 52 y 苷共兹5 兲 2 4 63. 共x 1兲2 共 y 3兲2 苷 52 65. center 共3, 0兲, radius 2 67. center 共7, 4兲, radius 3 1 1 3 5 69. center , 0 , radius 4 71. center , , radius 2 2 2 2

y

4

−4

4

冉冊冉冊冉冊

x

−4

73. 共x 1兲2 共 y 7兲2 苷 25

75. 共x 7兲2 共 y 11兲2 苷 121

77.

冉

y

2 −2

2

x

−2

−2

87. 共13, 5兲 89. 共7, 6兲 91. x 2 6x y 2 8y 苷 0 95. 共x 3兲2 共 y 3兲2 苷 32

y

85.

y

81.

2

2 3 x

y

冊

y

79.

3

83.

x

2

−2

x

2

x

93. 9x 2 25y 2 苷 225

2 x

Prepare for This Section (1.3), page 30 PS1. 4

PS2. 1 and 3

PS3. 3 and 3

PS4. 兹58

PS5. 4

PS6. decrease

Exercise Set 1.3, page 46 1. a. 5 b. 4 c. 1 d. 1 e. 3k 1 f. 3k 5 3. a. 兹5 b. 3 c. 3 d. 兹21 e. 兹r 2 2r 6 f. 兹c 2 5 1 1 5 1 1 5. a. b. c. d. 1 e. 2 f. 7. a. 1 b. 1 c. 1 d. 1 e. 1 f. 1 9. a. 11 b. 6 2 2 3 c 4 兩2 h兩 2 c. 3c 1 d. k 2k 10 11. Yes 13. No 15. No 17. Yes 19. No 21. Yes 23. Yes 25. Yes 27. all real numbers 29. all real numbers 31. {x 兩 x 苷 2其 33. 兵x 兩 x 7其 35. 兵x 兩 2 x 2其 37. 兵x 兩 x 4其 39.

41.

y

y 4

43.

y

4

2

2

x

2 −6 −4

47.

y

45.

int [10 2(2.3458) 0.5] 苷 2.35 10 2

−2

49.

2

−4

4 x

int [10 3(34.05622) 0.5] 苷 34.056 10 3

2 −2

4

6 x

−2

2 −2

−4

51.

int [10 4(0.08951) 0.5] 苷 0.0895 10 4

x

A3

Answers to Selected Exercises

55. a, b, and d. 57. decreasing on 共, 0兴; increasing on 关0, 兲 59. increasing on 共, 兲 61. decreasing on 共, 3兴; increasing on 关3, 0兴; decreasing on 关0, 3兴; increasing on 关3, 兲 63. constant on 共, 0兴; increasing on 关0, 兲 65. decreasing on 共, 0兴; constant on 关0, 1兴; increasing on 关1, 兲 67. g and F 69. a. w 苷 25 l b. A 苷 25l l 2 71. v共t兲苷 80,000 6500t, 0 t 10 73. a. C共x兲苷 2000 22.80x b. R共x兲苷 37.00x c. P共x兲苷 14.20x 2000 75. h 苷 15 5r 77. d 苷兹9t 2 2500 79. d 苷兹共45 8t兲2 共6t兲2 1 1 5 81. a. A共x兲苷 x 2 x 25 b. 25, 17.27, 14.09, 15.46, 21.37, 31.83 4p 16 2 c. 关0, 20兴 83. a. L共x兲苷兹900 x 2 兹400 共40 x兲2 b. 74.72, 67.68, 64.34, 64.79, 70 c. 关0, 40兴 85. 275, 375, 385, 390, 394

b. C(w) Cost (in dollors)

53. a. $1.07

2

1

1

2

3

4

冉

w

5

Weight (in ounces)

87. c 苷 2 or c 苷 3

89. 1 is not in the range of f.

冊

2

91.

1

93. −4.7

−4.7

4.7

4.7

−5

−5 2

95.

97. 4

−4.7

99. 2

101. a. 36

b. 13

c. 12

e. 13k 2

d. 30

f. 8k 11

103. 4兹21

4.7

−4

105. 1, 3

y

107.

4

4 x

−4

Prepare for This Section (1.4), page 53 PS1. The graph of g is one unit above the graph of f.

PS4. 3, 1, 3, 3, 1

PS5. 共0, b兲

PS6. 共0, 0兲

Exercise Set 1.4, page 64 1.

y

C(−5, 3)

A(−5, −3)

y

3.

B(5, 3)

R(−2, 3)

A(2, 3)

x

P(5, −3)

y

5.

B(−2, −3)

C(2, −3)

y

7.

B(−4, 5)

y

9. 2

−2

x

x

x

3

x

T(−4, −5)

17. a. Yes 33.

−3 x

A(4, −5)

b. Yes y

19. a. Yes 35.

b. Yes

21. a. Yes 37.

y

b. Yes y (0, 8)

23. No

3 (1, 0) (− 1, 0)

x (0, − 1)

(− 1, 0)

y

−3

C(4, 5)

−3

13. a. No b. Yes 15. a. No b. No y 27. Yes 29. Yes 31.

11.

(1, 0) x (0, 0)

3

x

3

(4, 0)

x

25. Yes

A4

Answers to Selected Exercises y

39.

y

41. 5

43. even 57. a., b.

x

−2

45. odd

47. even

f(x) + 3

51. even

53. even b.

y

f(x − 3)

−3

55. neither y

2

2 y = f(x + 2)

x (4, 0)

(0, 0)

49. even 59. a.

y

x

intercepts: (a, 0), a 0

61. a. 共5, 5兲, 共3, 2兲, 共2, 0兲

b. 共2, 6兲, 共0, 1兲, 共1, 1兲

−4 −2

y

63. a.

y = f(x) + 2

4x

2

−4 −2

y 4

b.

2

2 y = f(−x) −4 −2

4x

−2

−4

4x

2

y = −f(x) −4

65. a. 共1, 3兲, 共2, 4兲

b. 共1, 3兲, 共2, 4兲

y 4

67. a., b.

y

69. F(−x)

2

6

− 4 −2

−2

−F(x)

4

4 x

2

−4

−6 −4

6 x

−2 −4 −6

y

71. a.

y

b.

f (2x) 1 −2 −1

−6

1 2

x

6

−6

y

73. a.

1

f 3x

2

−2 −1

2

y

b.

2

2

−2π

−π

π

2π

x

−2π

−π

π

−2

6

y = h( 1 x) 2

3

y = √x + 3

7

77.

y = √x 3

y = √x − 1 5

−4

7

79.

J(x) for

3

−5

x

2π

−2

y = h(2x)

75.

x

6

y = 1 x2 2

c=2

y = x2

c=0 c = −1

y = 2x 2

6 −4 −3

−4 5

81.

4

83. a.

−1

y

b.

y

6 y = 1 (|x −1| − |x|) −5

5

y = |x −1| − |x| y = 4 (|x −1| − |x|)

−5

2

4

4 2

2

2

4

6

x

4

6

x

2

4x

Answers to Selected Exercises c.

y

85. a. f 共x兲苷

2 1 共x 1兲2 1

b. f 共x兲苷

2 共x 2兲2 1

12 10 8 6 4 2

2

4

6

x

Prepare for This Section (1.5), page 69 PS1. 2

PS2. 90

PS3. 18a2 15a 2

PS4. 2h2 3h

PS5. all real numbers except x 苷 1

PS6. [4, )

Exercise Set 1.5, page 77 1. f 共x兲 g共x兲苷 x 2 x 12, Domain is the set of all real numbers. f 共x兲 g共x兲苷 x 2 3x 18, Domain is the set of all real numbers. f 共x兲 g共x兲苷 x 3 x 2 21x 45, Domain is the set of all real numbers. f 共x兲苷 x 5, Domain 兵x 兩 x 苷 3其 g共x兲 5. f 共x兲 g共x兲苷 x 3 2x 2 8x, Domain is the set of all real numbers. f 共x兲 g共x兲苷 x 3 2x 2 6x, Domain is the set of all real numbers. f 共x兲 g共x兲苷 x 4 2x 3 7x 2, Domain is the set of all real numbers. f 共x兲苷 x 2 2x 7, Domain 兵x 兩 x 苷 0其 g共x兲 9. f 共x兲 g共x兲苷兹x 3 x, Domain 兵x 兩 x 3其 f 共x兲 g共x兲苷兹x 3 x, Domain 兵x 兩 x 3其 f 共x兲 g共x兲苷 x兹x 3, Domain 兵x 兩 x 3其 f 共x兲兹x 3 苷 , Domain 兵x 兩 x 3其 g共x兲 x 9 384 13. 18 15. 17. 30 19. 12 21. 300 23. 4 125 35. 8x 4h

37. 共 g ⴰ f 兲共x兲苷 6x 3 共 f ⴰ g兲共x兲苷 6x 16

兹1 x 2 兩x兩 1 共 f ⴰ g兲共x兲苷 x1

45. 共 g ⴰ f 兲共x兲苷

61. 16c 2 4c 6

3. f 共x兲 g共x兲苷 3x 12, Domain is the set of all real numbers. f 共x兲 g共x兲苷 x 4, Domain is the set of all real numbers. f 共x兲 g共x兲苷 2x 2 16x 32, Domain is the set of all real numbers. f 共x兲苷 2, Domain 兵x 兩 x 苷 4其 g共x兲

7. f 共x兲 g共x兲苷 2x 2 7x 12, Domain is the set of all real numbers. f 共x兲 g共x兲苷 2x 2 x 2, Domain is the set of all real numbers. f 共x兲 g共x兲苷 8x 3 2x 2 41x 35, Domain is the set of all real numbers. f 共x兲 4x 7 5 苷 2 , Domain x 兩 x 苷 1, x 苷 g共x兲 2x 3x 5 2 11. f 共x兲 g共x兲苷兹4 x 2 2 x, Domain 兵x 兩 2 x 2其 f 共x兲 g共x兲苷兹4 x 2 2 x, Domain 兵x 兩 2 x 2其 f 共x兲 g共x兲苷共兹4 x 2 兲共2 x兲, Domain 兵x 兩 2 x 2其 f 共x兲兹4 x 2 苷 Domain 兵x 兩 2 x 2其 g共x兲 2x 5 1 25. 27. 29. 2 31. 2x h 33. 4x 2h 4 2 4

再

39. 共 g ⴰ f 兲共x兲苷 x 2 4x 1 共 f ⴰ g兲共x兲苷 x 2 8x 11

2兩5 x兩 3 3兩x兩共 f ⴰ g兲共x兲苷兩5x 2兩

47. 共 g ⴰ f 兲共x兲苷

63. 9k 4 36k 3 45k 2 18k 4

49. 66

51. 51

41. 共 g ⴰ f 兲共x兲苷 5x 3 10x 共 f ⴰ g兲共x兲苷 125x 3 10x

53. 4

55. 41

57.

3848 625

冎

1 5x x1 2 共 f ⴰ g兲共x兲苷 3x 4

43. 共 g ⴰ f 兲共x兲苷

59. 6 2兹3

65. a. A共t兲苷 p共1.5t兲2, A共2兲苷 9p square feet ⬇ 28.27 square feet

b. V共t兲苷 2.25pt 3, V共3兲苷 60.75p cubic feet ⬇ 190.85 cubic feet

67. a. d共t兲苷兹共48 t兲2 42

b. s共35兲苷 13 feet, d共35兲 ⬇ 12.37 feet

A5

A6

Answers to Selected Exercises

69. 共Y ⴰ F 兲共x兲 converts x inches to yards. c. 49.7 d. 30.8 e. 16.4 f. 0

71. a. 99.8; this is identical to the slope of the line through 共0, C共0兲兲 and 共1, C共1兲兲.

b. 156.2

Prepare for This Section (1.6), page 81 2 PS1. y 苷 x 3 5

PS2. y 苷

1 x1

PS3. 1

PS4. 共3, 7兲

PS6. 兵x兩x 2其

PS5. all real numbers

Exercise Set 1.6, page 90 1. 3

3. 3

5. 3

7. range

y

9. Yes

(0, −4)

y

17. Yes

19. Yes

4

(−5, 6) (−3, 0)

8 x

−8

(0, −4) 8

(− 2, − 8)

15. No

y

13. Yes

8

(3, 2) −8

y

11. Yes (6, 8)

8

−8

21. No

23. Yes

25. Yes

x

4 x

−4 −4

(4, −6)

27. 兵共1, 3兲, 共2, 2兲, 共5, 1兲, 共7, 4兲其

8

8 x

−8 −8

29. 兵共1, 0兲, 共2, 1兲, 共4, 2兲, 共8, 3兲, 共16, 4兲其 39. f 1共x兲苷

x1 ,x苷1 1x

31. f 1共x兲苷

1 x2 2

41. f 1共x兲苷兹x 1, x 1

47. f 1共x兲苷兹x 5 2, x 5

33. f 1共x兲苷

1 7 x 3 3

35. f 1共x兲苷

43. f 1共x兲苷 x 2 2, x 0

1 5 x 2 2

37. f 1共x兲苷

x ,x苷2 x2

45. f 1共x兲苷兹x 4 2, x 4

3

49. V 1共x兲苷兹 x. V 1 finds the length of a side of a cube given the volume.

51. f 1共x兲苷

9 x 32; 5

1 x 12 2 1 55. E 共s兲苷 20s 50,000. The executive can determine the value of the software that must be sold in order to achieve a given monthly income. 57. a. p(10) ⬇ 0.12 苷 12%; p(30) ⬇ 0.71 苷 71% b. The graph of p, for 1 n 60, is an increasing function. Thus p (with 1 n 60) has an inverse that is a function. c. Answers will vary. 59. a. 25 47 71 67 47 59 53 71 33 47 43 27 63 47 53 39 b. PHONE HOME c. Answers will vary. 61. Because the function is increasing and 4 is between 2 and 5, c must be between 7 and 12. 63. between 2 and 5 1 b 65. between 3 and 7 67. slope: ; y-intercept: 0, 69. The reflection of f across the line given by y 苷 x yields f. Thus f is its own inverse. m m 71. Yes 73. No f 1共x兲 is used to convert x degrees Celsius to its equivalent Fahrenheit temperature.

53. s1(x) 苷

冉冊

Prepare for This Section (1.7), page 95 3 PS1. slope: ; y-intercept: 4 4

3 PS2. slope: ; y-intercept: 3 4

PS3. y 苷 0.45x 2.3

PS4. 19

PS5. 69

PS6. 3

Exercise Set 1.7, page 103 1. no linear relationship 3. linear 5. Figure A 7. y 苷 2.00862069x 0.5603448276 9. y 苷 0.7231182796x 9.233870968 11. y 苷 2.222641509x 7.364150943 13. y 苷 1.095779221x 2 2.69642857x 1.136363636 15. y 苷 0.2987274717x 2 3.20998141x 3.416463667 17. a. y 苷 23.55706665x 24.4271215 b. 1248 centimeters 19. a. y 苷 0.1094224924x 0.7978723404 b. 4.3 meters per second 21. a. y 苷 0.1628623408x 0.6875682232 b. 25 23. No, because the linear correlation coefficient is close to 0. 25. a. Yes, there is a strong linear correlation. b. y 苷 0.9033088235x 78.62573529 c. 56 years 27. a. positively b. 1098 calories 29. y 苷 0.6328671329x 2 33.61608392x 379.4405594 31. a. y 苷 0.0165034965x 2 1.366713287x 5.685314685 b. 32.8 miles per gallon 33. a. 5-pound: s 苷 0.6130952381t 2 0.0714285714t 0.1071428571 10-pound: s 苷 0.6091269841t 2 0.0011904762t 0.3 15-pound: s 苷 0.5922619048t 2 0.3571428571t 1.520833333 b. All the regression equations are approximately the same. Therefore, the equations of motion of the three masses are the same.

Answers to Selected Exercises

A7

Chapter 1 Assessing Concepts, page 111 1. a, c, d, and e 8. 共3, 6兲

2. No

9. 共3, 4兲

4. 兩x 2兩 3

3. 3

5. It is the slope of the line between 共a, f共a兲兲 and 共b, f共b兲兲.

6. 共3, 2兲

7. 共7, 3兲

10. Yes. The slope of the regression line in negative.

Chapter 1 Review Exercises, page 112 1.

9 [1.1] 4

2. 4 [1.1]

9. c 6 [1.1] 15. 10 [1.2]

[1.4]

y

4.

17. 5 [1.2] 24. a.

a.

c.

18. 7兹5 [1.2]

5. 3, 6 [1.1] 12.

19. 兹181 [1.2]

4 x

2

20. 4兹5 [1.2]

y 4

14. 1 x

冊

1 , 10 [1.2] 2 [1.4]

5 [1.1] 3

22. 共2, 2兲 [1.2]

2 4 x

2

−2

−4

冉

21.

c.

y 4

−4

3 兹41 [1.1] 4

8.

13. 共, 1兲共4, 兲 [1.1]

2

−4 −2 −2

−4

1 兹13 [1.1] 6

7.

1 x 1 [1.1] 2

2 4x

1 , 4 [1.1] 2

6.

b.

y 4

b. −4

2 [1.1] 3

11. 共, 3兴关4, 兲 [1.1]

10. a 1 [1.1] 16. 3兹31 [1.2]

23.

3. 2 [1.1]

−4 −2 −2

−4

4 x

2

−4

25. y-axis [1.4] 26. x-axis [1.4] 27. origin [1.4] 28. x-axis, y-axis, origin [1.4] 29. x-axis, y-axis, origin [1.4] 30. origin [1.4] 31. x-axis, y-axis, origin [1.4] 32. origin [1.4] 33. center 共3, 4兲, radius 9 [1.2] 34. center 共5, 2兲, radius 苷 3 [1.2] 35. 共x 2兲2 共 y 3兲2 苷 5 2 [1.2] 36. 共x 5兲2 共 y 1兲2 苷 64 [1.2] 37. a. 2 b. 10 c. 3t 2 4t 5 d. 3x 2 6xh 3h2 4x 4h 5 e. 9t 2 12t 15 f. 27t 2 12t 5 [1.3] 38. a. 兹55 b. 兹39 c. 0 d. 兹64 x2 2 2 2 2 e. 2兹64 t f. 2兹16 t [1.3] 39. a. 5 b. 11 c. x 12x 32 d. x 4x 8 [1.5] 40. a. 79 b. 56 c. 2x 2 4x 9 44. x 6 [1.3] 47. y

d. 2 x 2 6 [1.5] 45. 关5, 5兴 [1.3] [1.3] 48.

41. 8x 4h 3 [1.5] 42. 3x2 3xh h2 1 [1.5] 46. all real numbers except 3 and 5 [1.3] y y [1.3] 49. [1.3]

3

increasing on [3, ) decreasing on (, 3]

4

−2

−3

x

y

[1.3] 6

52.

x

increasing on (, )

[1.3/1.4] 56.

6

−2

2

4 x

54.

x

a. domain 共, 兲 range 兵y 兩 y 4其 b. even

[1.3/1.4]

y 4

1

−2

x

−4 −2

2

4 x

−4

a. domain 共, 兲 range 兵y 兩 y 4其 b. even

[1.3/1.4] 57.

y

y

[1.3/1.4]

a. domain 共, 兲 range 共, 兲 b. neither 58.

2

4 x

[1.3/1.4]

y 4

2

2

3

[1.3/1.4]

−4

−4 −2 −2 −3

y

2

increasing on (, )

y

53.

x

2

constant on 关n, n 1兲, where n is an integer

2 −4

55.

x

increasing on 关2, 2兴 constant on 共, 2兴关2, 兲

[1.3]

y 4

3

−2

decreasing on 共, 0兴 increasing on 关0, 兲

−2

[1.3]

4

2

x

−4

51.

y

50.

3

4 5

43. (, ) [1.3]

2 2 x

−4 −2

2

4 x

−4

a. domain 关4, 4兴 range 关0, 4兴 b. even

a. domain 共, 兲 range 共, 兲 b. odd

a. domain 共, 兲 range: even integers b. neither

A8

Answers to Selected Exercises

59. f 共x兲 g共x兲苷 x 2 x 6, domain: 共, 兲 60. f 共x兲 g共x兲苷 x 2 x 12, domain: 共, 兲 f 共x兲 g共x兲苷 x 3 3x 2 9x 27, domain: 共, 兲 f 共x兲苷 x 3, domain 兵x 兩 x 苷 3其 [1.5] g共x兲 x4 63. Yes [1.6] 64. No [1.6] 65. f 1共x兲苷 [1.6] 3

f 共x兲 g共x兲苷 x 3 x 2 2x 12, domain: 共, 兲 61. Yes [1.6] 62. Yes [1.6] f 共x兲 g共x兲苷 x 3 x 2 2x 4, domain: 共, 兲 f 共x兲 g共x兲苷 x 5 2x 4 4x 3 8x 2 16x 32, domain: 共, 兲 f 共x兲苷 x 2, domain: 共, 兲 [1.5] g共x兲 1 3 1 66. g1共x兲苷 x [1.6] 67. h1共x兲苷 2x 4 [1.6] 68. k1共x兲苷 [1.6] 2 2 x

y 4

y

y 4

g−1

h

2

f −1 x

4

f

h

g

−4

x

4

4 x

− 4 −2 −2

y 4 k 2 k −1

−1

−4 −2 −2

2

4 x

−4

69. t ⬇ 3.7 seconds [1.3] 70. a. 150 feet b. 525 feet [1.3] 71. a. y 苷 0.018024687x 0.00050045744 b. Yes. r ⬇ 0.999, which is very close to 1. c. 1.8 seconds [1.7] 72. a. y 苷 0.0047952048x2 1.756843157x 180.4065934 b. The graph of the regression equation never crosses the x-axis. Therefore, the model predicts that the can will never be empty. c. A regression model only approximates a situation. [1.7]

Chapter 1 Quantitative Reasoning Exercises, page 114 QR1. a. 3

b. 2

c. 52

QR2. Factor the modulus, m.

QR3. Answer will vary.

Chapter 1 Test, page 116 1 1 [1.1] 2. x 1 [1.1] 3. , 2 [1.1] 2 2 y 8. 共0, 兹2 兲, 共0, 兹2 兲, 共4, 0兲

4.

1.

2 , 1 [1.1] 3 [1.2] 9.

冉

2

y

13.

[1.3/1.4]

x

2

−2

[1.4] 4

2 −2 −2

2

x

−4

increasing on 共, 2兴 decreasing on 关2, 兲 18. x 2兹x 2 1 [1.5]

−2 −2

y

14.

19. f 1共x兲苷

冊

2 共2, 兲 [1.1] 6. 兹85 [1.2] 7. midpoint 共1, 1兲; length 2 兹13 [1.2] 5 [1.2] 10. center 共2, 1兲; radius 3 [1.2] y 11. 4 [1.3] 12. domain 兵x 兩 x 4 or x 4其 [1.3] 2

5. ,

2

x

15. b [1.4]

16. x 2 x 3;

y = f(x)

x2 1 , x 苷 2 [1.5] x2

17. 2x h [1.5]

4 x −2 y = −f (x + 2) − 1

x [1.6] 1x

20. a. y 苷 7.98245614x 767.122807

b. 56.7 calories [1.7]

Exercise Set 2.1, page 130 1. 75 , 165

15. 105 , Quadrant II 17. 11 31. 33. 35. 6 2 12 57. 2.32

3 3 1, 1 9. , , 11. 13. 250 , Quadrant III 2 4 4 10 5 296 , Quadrant IV 19. 24 33 36 21. 64 9 28.8 23. 3 24 7.2 25. 25.42 27. 183.56 29. 211.78 7 13 p 37. 39. 41. 43. 420 45. 36 47. 30 49. 67.5 51. 660 53. 75 55. 85.94 3 4 20 5 4, 229.18 63. 2.38, 136.63 65. 6.28 inches 67. 18.33 centimeters 69. 3 71. radians or 75 12

3. 19 45 , 109 45

59. 472.69

61.

5. 33 26 45, 123 26 45

7.

Answers to Selected Exercises 5 10 radian per second ⬇ 0.105 radian per second 75. radians per second ⬇ 5.24 radians per second 77. radians per 30 3 9 second ⬇ 3.49 radians per second 79. 40 mph 81. 1885 feet 83. 6.9 mph 85. 840,000 miles 87. a. 3.9 radians per hour b. 27,300 kilometers per hour 89. a. B b. Both points have the same linear velocity. 91. a. 1.15 statute miles b. 10% 93. 13 square inches 95. 4680 square centimeters 97. 1780 miles 73.

Prepare for This Section (2.2), page 134 PS1.

兹3 3

PS2. 兹2

PS3. 2

PS4.

兹3 3

PS5. 3.54

PS6. 10.39

Exercise Set 2.2, page 142 5兹29 12 13 4 7 兹29 csc 苷 csc 苷 3. sin 苷 5. sin 苷 csc 苷 5 13 12 7 4 29 5 13 7兹33 2兹29 兹33 兹29 cos 苷 sec 苷 cos 苷 sec 苷 cos 苷 sec 苷 13 5 7 33 29 2 12 5 4兹33 5 2 兹33 tan 苷 cot 苷 tan 苷 cot 苷 tan 苷 cot 苷 5 12 33 4 2 5 2兹3 6兹61 兹21 兹21 兹3 兹61 csc 苷 csc 苷 csc 苷 7. sin 苷 9. sin 苷 11. sin 苷 7 3 2 3 61 6 2兹7 1 5兹61 兹7 兹61 sec 苷 sec 苷 2 cos 苷 cos 苷 cos 苷 sec 苷 7 2 2 61 5 2兹3 6 5 兹3 兹3 tan 苷兹3 cot 苷 tan 苷 tan 苷 cot 苷 cot 苷 2 3 3 5 6 3 5 3 兹3 3 4 3 12 13 3 3兹2 2兹3 13. 15. 17. 19. 21. 23. 25. 兹2 27. 29. 31. 兹3 33. 35. 4 5 4 13 5 2 4 4 6 3 37. 2兹2 兹3 39. 0.6249 41. 0.4488 43. 0.8221 45. 1.0053 47. 0.4816 49. 1.0729 51. 9.5 feet 53. 92.9 inches 55. 5.1 feet 57. 1.7 miles 59. 74.6 feet 61. 686,000,000 kilometers 65. 612 feet 67. 560 feet 69. a. 559 feet b. 193 feet 71. 兹27 meters⬇5.2 meters 73. ⬇8.5 feet 1. sin 苷

Prepare for This Section (2.3), page 147 PS1.

4 3

PS2.

兹5 2

PS3. 60

PS4.

5

PS5.

PS6. 兹34

Exercise Set 2.3, page 153 3兹13 3兹13 5兹89 兹13 兹13 兹89 csc 苷 csc 苷 3. sin 苷 csc 苷 5. sin 苷 13 3 13 3 89 5 2兹13 2兹13 8兹89 兹13 兹13 兹89 cos 苷 sec 苷 cos 苷 sec 苷 cos 苷 sec 苷 13 2 13 2 89 8 3 2 3 2 5 8 tan 苷 cot 苷 tan 苷 cot 苷 tan 苷 cot 苷 2 3 2 3 8 5 7. sin 苷 0 csc is undefined. 9. 0 11. 0 13. 1 15. 0 17. undefined 19. 1 21. Quadrant I 23. Quadrant IV cos 苷 1 sec 苷 1 tan 苷 0 cot is undefined. 2兹3 8 兹3 兹3 兹3 25. Quadrant III 27. 29. 1 31. 33. 35. 37. 20 39. 9 41. 43. 45. 34 3 3 3 3 5 3 2兹3 兹2 兹2 47. 65 49. 51. 1 53. 55. 57. 兹2 59. cot 540 is undefined. 61. 0.798636 63. 0.438371 2 3 2 3 65. 1.26902 67. 0.587785 69. 1.70130 71. 3.85522 73. 0 75. 1 77. 79. 1 81. 30 , 150 83. 150 , 210 2 3 7 5 11 2 85. 225 , 315 87. , 89. , 91. , 4 4 6 6 3 3 1. sin 苷

A9

A10

Answers to Selected Exercises

Prepare for This Section (2.4), page 155 PS1. Yes

PS2. Yes

PS4. 2

PS3. No

PS5. even function

Exercise Set 2.4, page 166 1.

冉冊冉

17. 35. 63. 83. 97.

冊冉

冊冉

PS6. neither

冊

冉

冊

1 1 1 1 1 兹3 1 兹3 兹3 兹3 兹3 兹3 , , , , 3. 5. 7. 9. 共1, 0兲 11. , 13. 15. 2 2 2 2 2 2 2 2 2 2 3 2 2兹3 2兹3 19. 1 21. 23. 0.9391 25. 1.1528 27. 0.2679 29. 0.8090 31. 48.0889 33. a. 0.9 b. 0.4 3 3 a. 0.8 b. 0.6 37. 0.4, 2.7 39. 3.4, 6.0 41. odd 43. neither 45. even 47. odd 49. 2p 51. p 53. 2p 61. sin t 71. 2 csc2 t 73. csc2 t 75. 1 77. 兹1 cos2 t 79. 兹1 cot 2 t 81. 750 miles sec t 65. tan2 t 67. cot t 69. cos2 t sin2 t 85. csc t sec t 87. 1 2 sin t sin2 t 89. 1 2 sin t cos t 91. cos2 t 93. 2 csc t 95. 共cos t sin t兲共cos t sin t兲 cos t 兹2 兹3 99. 共2 sin t 1兲共sin t 1兲 101. 103. 共tan t 2)共tan t 3) 2 3

Prepare for This Section (2.5), page 169 PS2. 0.7 1 y-axis by a factor of . 2

PS3. Reflect the graph of y 苷 f 共x兲 across the x-axis.

PS1. 0.7

PS5. 6

PS4. Compress each point on the graph of y 苷 f 共x兲 toward the

PS6. 5

Exercise Set 2.5, page 177 1. 2, 2 19.

3. 1,

5.

1 ,1 2

y

7. 2, 4

9.

1 , 2 2

y

21.

1

11. 1, 8 y 4

23.

3

15. 3, 3

13. 2, 6

17. 4.7, 2.5

y 4

25.

y

27.

1

2 2π x

π

π 2π

x

−2

π

2π x

π

2π

x

−4

x

2π 3

−1

−4

29.

y

31. 2π 3

1

4π 3

x

−1

39.

y 2

y

33.

1

41.

−1

1

2

3

4

y

43.

1

49.

2

4 x −2

y

51.

−2

2π

y

4π x −2

1

y

3 x

53.

x

π 3

2π 3

x

−2

4π x

2π

47.

2 π

−2

x

55. 3π

3π

−2

y

π 2

y

2

2

y

2

y

37.

−4

45.

−2

2

2

x

2

2 −2

y 4

35.

−1

2 x

y 1

2π 3

−2

2 x

6π x −3

x

57. y 苷 cos 2x

y

1

4π 3

6π x

A11

Answers to Selected Exercises 59. y 苷 2 sin

2 x 3

61. y 苷 2 cos x

y

65.

63. a. V 苷 4 sin pt, 0 t 8 milliseconds y

67.

6π x

−3π

−2π

4π x

2π

−2

11

69.

y2 = 2 cos x

2

2

1 cycle per millisecond 2

b.

−1

20

y1 = 2 cos x 2 −11

20

71.

75. y 苷 2 sin

73. y 3

−1

77. y 苷 2.5 sin

5p x 8

2

20

1 π

2π

3π

−20

maximum 苷 e, minimum 苷 79. y 苷 3 cos

2 x 3

4π

x

1 ⬇ 0.3679, period 苷 2 e

4 x 5

Prepare for This Section (2.6), page 180 PS3. Stretch each point on the graph of y 苷 f 共x兲 away from the x-axis by a factor of 2. 4 PS4. Shift the graph of y 苷 f 共x兲 2 units to the right and 3 units up. PS5. 2 PS6. 3

PS1. 1.7

PS2. 0.6

Exercise Set 2.6, page 189 1.

k, k an integer 2 y

23.

3.

k, k an integer 2 y

25.

3 −π 2

33.

5. 2 27.

3

−3

π

−3

y

35.

11.

y

2 3

x

37.

x

2π

−2

y

2π

−3

y

y

43.

−2 −π

π

2π x

19. 5

21. 8.5

31.

y

2 2π x

π

y

−π

41.

−2

π

y

3 π

x

x

2

−1

−2

4

y

47.

49.

2

4 x

−π

−3

1 4

x

y

2

3

2

17. 1

2

y

45.

15. 8

y

−1

39.

−1

6π x

3π

−2

3

1

2

x

13.

29.

1

3

−2π

9. 2

2

x

π 2

7.

2 π

x

−2

2 x

x

A12

Answers to Selected Exercises

51. y 苷 cot

3 x 2

53. y 苷 csc

2 x 3

3 x 4

55. y 苷 sec

4

57.

59.

−1.5π

6

1.5π −π

2π

−4

61. a. h 苷 1.4 tan x

b. d 苷 1.4 sec x

20

c.

d

h π/2

0 0

Xscl = π/8, Yscl = 1

−2

d. The graph of d is above the graph of h, but the distance between p the graphs approaches 0 as x approaches . 2 8 63. y 苷 tan 3x 65. y 苷 sec x 3 4 67. y 苷 cot 69. y 苷 csc x x 2 3

Prepare for This Section (2.7), page 192 PS1. amplitude 2, period

PS2. amplitude

2 , period 6 3

PS3. amplitude 4, period 1

PS5. 3

PS4. 2

PS6. y-axis

Exercise Set 2.7, page 199 , 2 2

1. 2,

3. 1,

, 8

17. y 1

5. 4,

5π 2

5 2 2 , , 4 3 3

y 2

, 8 2

11. 3, 6 y

4π 3

27.

4

3

y

29.

π 4

5π 4

33. y

3π 2

35. y

9π 2

37.

−1

π 4

15π x 2

31.

−2

y 6

39. π

8π x

2π

41.

3

5

7

x

1

π 2

5π 2

1 x

−π

π x

y 1

x

2π

x

x

−4

49. y

π

π

−2

−4

47.

7π 12

y

2 1

y

3π 12

4

x

2π x

45.

y

2

−2

y

2π x

43. y

x

9π 4

y

1

1 1

y

1

x

π 4

15. 12, 4

2

−1

x

7π 4

, 16

23.

−1

− 3π

1 −π 4

13.

1 10π x

−1

− 2π 3

9.

21. 1

x

25.

7.

y

19. π 2

, 3 4

51.

y

1 2π

4π x

π

y = x − sin x y=x y = −sin x

π

2π

x

A13

Answers to Selected Exercises

y

53.

55. y=x

π

y

1

y = x + sin 2x y = sin 2x x π

−1

63. a. 7.5 months, 12 months

冉

57. y 苷 sin 2x

y = sin x + cos x y = sin x 2π

y = cos x

Sales in hundreds of suits

冊

冉冊

59. y 苷 csc

x 2

冉冊

61. y 苷 sec x

2

x

y

b.

3

c. August

65. ⬇ 20 parts per million

10 7

S

6 7.5

y1

−2

6

−4

t

10

y2

Time in months

67. s 苷 7 cos 10t 5

t, t in seconds 5

69. s 苷 400 tan

y 12

71. y 苷 3 cos

t 9, 12 feet at 6:00 P.M. 6

2

73.

y 0

4π

400 −2

t 0.2 min

3

75.

−2

6

77.

0

10 t

5

−400

4π

0

−3

4π

冉

2 3

冊

冉

85. y 苷 tan

1.5

81.

0

−6

83. y 苷 2 sin 2x

25

79.

8π

−0.2

−25

x 2 4

冊

87. cos2 x 2

−1.5

1.5

89.

4

91.

6

−2π −4π − 0.5

The graph above does not show that the function is undefined at x 苷 0.

Prepare for This Section (2.8), page 203 PS1.

3 2

PS2.

5 2

PS3. 4

PS4. 3

PS5. 4

PS6. y 苷 4 cos x

2π

4π

−6

A14

Answers to Selected Exercises

Exercise Set 2.8, page 208 1. 2, ,

1

3. 3, 3,

1 3

9. y 苷 4 cos 3t

5. 4, 2, 11. y 苷

y

1 2

7.

3 4 cos t 2 3

1 3 , 4, 4 4

y

4

19. y 苷

1 cos 4t 2

17. y 苷 2 sin 2t

y

y

y

2

1

2

1 2

15. y 苷 sin t

13. y 苷 2 sin 2t

4

2π

t

21. y 苷 2.5 cos t

1 second 196 35. a. 10 b. 71.0 seconds 37. a. 10 43. Yes 45. Yes

1 2 cos t 2 3

t

t

1

t

1 t , 2 feet; y 苷 2 cos 4 2 p b. The amplitude needs to increase. 31. h 苷 37 cos 33. a. 3 b. 59.8 seconds t 41 22.5 b. 9.1 seconds 39. a. 10 b. 6.1 seconds 41. The new period is 3 times the original period.

23. y 苷

29. a. 196 cycles per second;

π

t

3

25. y 苷 4 cos 4t

27. 4,

冉冊

Chapter 2 Assessing Concepts, page 212 1. True

2. False

3. False

4. True

9. all real numbers except multiples of p

5.

p 4

10. x 苷

6. (0, 1)

7.

8 3

8. Shift the graph of y1 to the left

p units. 2

p 3p and x 苷 2 2

Chapter 2 Review Exercises, page 212 7 [2.1] 5. 3.93 meters [2.1] 6. 0.3 [2.1] 4 3兹10 2 3兹5 兹10 兹5 兹5 7. 55 radians per second [2.1] 8. [2.2] 9. [2.2] 10. [2.2] 11. [2.2] 12. sin 苷 csc 苷 [2.3] 3 2 3 5 10 3 兹10 cos 苷 sec 苷兹10 10 1 tan 苷 3 cot 苷 3 2兹3 1 1 兹3 13. a. b. 1 c. 1 d. [2.3] 14. a. 0.5446 b. 0.5365 c. 3.2361 d. 3.0777 [2.3] 15. a. b. [2.3] 3 2 2 3 2兹3 1 兹2 兹3 兹2 兹2 16. a. b. 2 [2.3] 17. a. b. 1 [2.3] 18. a. 共1, 0兲 b. , c. , d. 共1, 0兲 [2.4] 3 2 2 2 2 2 19. even [2.4] 22. sec2 [2.4] 23. tan [2.4] 24. sin [2.4] 25. tan2 [2.4] 26. csc2 [2.4] 27. 0 [2.4] 28. 3, , [2.5] 2 2 3 29. no amplitude, , 0 [2.6] 30. 2, , [2.5] 31. 1, , [2.5] 32. no amplitude, , [2.6] 33. no amplitude, 2, [2.6] 3 3 9 3 2 8 4 1. complement measures 25 ; supplement measures 115 . [2.1]

2. 80 [2.3]

3. 114.59 [2.1]

冉

34.

[2.5]

y 2

2

35.

[2.5]

y 1

x

冊冉

3π

x

冊

[2.5]

y

2 π

−1

36.

4.

2π 3

4π 3

37.

[2.7]

y 1

x

π −1

3π

x

A15

Answers to Selected Exercises 38.

[2.7]

y

39.

1

40.

7π x

[2.7]

y

43.

π 3

2π 3

−π

x

π

[2.7]

y

44.

π

[2.7]

y

1 π 2

45.

46.

y

π x

−π 8 −3

[2.7]

47.

3π 8

7π 8

π 6

x

[2.7]

y 5

π 2

48.

π −3 4 −6

[2.7]

y

5π 4

49.

−1

2π x

π

[2.7]

y

x

9π 4

[2.7]

y 1

4

π 2

x

−6 π 3

50.

x

7π 6

π 6

−4

x

[2.7]

y 6

x

π

π

2

1

π 2

[2.6]

y

x

3

1

41.

3

−3

8

8

[2.6]

y

1

−π

42.

[2.7]

y 3

51. y

[2.7]

2π 3

π

52. 0.089 mile [2.2]

x

53. 12.3 feet [2.2]

54. 1.7 feet per second [2.1]

2

2 π

x

−2

55. 46 feet [2.2]

56. 2.5,

25 , [2.8] 25

π 2

2π x

π

57. amplitude 苷 0.5, f 苷

1 , p 苷 , y 苷 0.5 cos 2t [2.8]

58. 7.2 seconds [2.8]

Chapter 2 Quantitative Reasoning Exercises, page 214 QR1. a. 6p

b. 4

c. 2p

d. 24p

f. 4p

e. 7.5

QR2. 15 seconds

QR3. 11.25 seconds

QR4. 54 seconds

Chapter 2 Test, page 214 兹58 [2.1] 3. 13.1 centimeters [2.1] 4. 12 radians兾second [2.1] 5. 80 centimeters兾second [2.1] 6. [2.2] 12 7 1 p 兹3 6 兹3 7. 1.0864 [2.2] 8. [2.3] 9. , [2.4] 10. sin2 t [2.4] 11. [2.6] 12. amplitude 3, period , phase shift [2.7] 6 2 2 3 4 1 13. period 3, phase shift [2.7] 14. y [2.5] 15. y [2.6] 2 4 4 1.

5 [2.1] 6

2.

冉

冊

2

2 −2 −4

π

2π

3π

4π x

−2 −4

π

2π

3π

4π

x

A16

Answers to Selected Exercises

16. Shift the graph of y 苷 2 sin 2x,

p units to the right and 1 unit down. [2.7] 4

17.

[2.7]

y 4

2 π

−2

18.

2

π

2π

3π

3π

4π x

2 t [2.8] 5

20. y 苷 13 sin

19. 25.5 meters [2.2]

[2.7]

y 4

2π

4π x

−2

Cumulative Review Exercises, page 215 3x 兹3 [1.2] 3. y-intercept 共0, 9兲, x-intercepts 共3, 0兲 and 共3, 0兲 [1.2] 4. odd function [1.4] 5. f 1共x兲苷 [1.6] 2 2x 1 6. 共, 4兲共4, 兲 [1.3] 7. 2, 3 [1.1] 8. Shift the graph of y 苷 f 共x兲 horizontally 3 units to the right. [1.4] 9. Reflect the graph of y 苷 f 共x兲 5 5 兹3 1 across the y-axis. [1.4] 10. [2.1] 11. 225° [2.1] 12. 1 [2.3] 13. [2.2] 14. [2.2] 15. negative [2.3] 16. 30° [2.3] 3 2 4 3 17. [2.3] 18. 共, 兲 [2.5] 19. 关1, 1兴 [2.5] 20. [2.2] 3 5 1.

5兹2 [1.2]

2.

Exercise Set 3.1, page 222 57. identity side is

59. identity

61. identity

2 兹3 兹3 and the right side is . 2 2

63. not an identity

65. If x 苷

p , the left side is 2 and the right side is 1. 4

67. If x 苷 0 , the left

69. If x 苷 0, the left side is 1 and the right side is 1.

Prepare for This Section (3.2), page 225 PS1. Both function values equal equal.

1 . 2

PS2. Both function values equal

1 . 2

PS3. For each of the given values of u, the function values are

PS4. For each of the given values of u, the function values are equal.

PS5. Both function values equal

兹3 . 3

PS6. 0

Exercise Set 3.2, page 233 1.

兹6 兹2 4

21. cot 75 63 65 56 47. a. 65 39. a.

3.

兹6 兹2 4

23. csc 65 56 65 63 b. 65 b.

5. 2 兹3

25. sin 5x

7.

兹6 兹2 4

27. cos x

63 63 41. a. 16 65 16 c. 75. cos u 63 c.

b.

9.

29. sin 4x 56 65

77. tan u

c.

33 56

兹6 兹2 4

31. cos 2x 43. a.

79. sin u

77 85

11. 2 兹3 33. sin x b.

81. identity

84 85

13. 0

35. tan 7x c.

13 84

15.

1 2

37. a.

45. a.

33 65

17. 兹3

19. cos 48

77 84 c. 85 36 16 63 b. c. 65 16

77 85

b.

83. identity

Prepare for This Section (3.3), page 236 2 tan a PS4. For each of the given values of a, the function values are equal. 1 tan2 a PS5. Let a 苷 45 ; then the left side of the equation is 1, and the right side of the equation is 兹2. PS6. Let a 苷 60 ; then the left side of the 1 兹3 equation is , and the right side of the equation is . 2 4 PS1. 2 sin a cos a

PS2. cos2 a sin2 a

PS3.

Exercise Set 3.3, page 243 1. sin 4a

3. cos 10b

5. cos 6a

7. tan 6a

9. sin 2 苷

24 7 24 240 161 240 , cos 2 苷 , tan 2 苷 , cos 2 苷 , tan 2 苷 11. sin 2 苷 25 25 7 289 289 161

A17

Answers to Selected Exercises 13. sin 2 苷

336 527 336 , cos 2a 苷 , tan 2a 苷 625 625 527

17. sin 2a 苷

720 1519 720 , cos 2a 苷 , tan 2a 苷 1681 1681 1519

23. 35.

1 (1 cos 2x cos 4x + cos 2x cos 4x) 16

兹2 兹3

25.

19. 3(1 cos 2x)

兹2 兹3

a 2兹5 a 1 a 兹5 苷苷苷 , tan , cos 2 2 2 2 5 5 2 1 ⬇ 2.61 b. a 苷 2 sin1 91. a. M 兹2 兹2 41. sin

43. sin

冉冊

1 (3 + 4 cos 2x + cos 4x) 8

21.

27. 兹2 1

2

a 5兹26 a a 兹26 37. sin 苷 , cos 苷 , tan 苷5 2 26 2 26 2

2

240 161 240 , cos 2a 苷 , tan 2a 苷 289 289 161

15. sin 2a 苷

29.

兹2 兹2

31.

2

兹2 兹2

33.

2

兹2 兹2 2

a 5兹34 a 3兹34 5 a 39. sin 苷 , cos 苷 , tan 苷 2 34 2 34 2 3

a 7兹2 a 1 a 兹2 苷苷苷 , tan , cos 2 2 7 2 10 10

c. a decreases.

93. identity

95. identity

Prepare for This Section (3.4), page 246 PS1. sin a cos b

PS2. cos a cos b

PS3. Both function values equal

1 . 2

PS4. sin x cos x

PS5. Answers will vary.

PS6. 2

Exercise Set 3.4, page 251 1 兹3 2 15. 4 4 3 1 2 sin 3u cos u 19. 2 cos 2u cos u 21. 2 sin 4u sin 2u 23. 2 cos 4u cos 3u 25. 2 sin 7u cos 2u 27. 2 sin u sin u 2 2 3 u 5 1 兹2 2 sin u sin sin共x 45 兲 31. 2 cos u sin u 49. y 苷兹2 sin共x 135 兲 51. y 苷 sin共x 60 兲 53. y 苷 4 4 12 12 2 3p p 2p y 苷 3兹2 sin共x 135 兲 57. y 苷 p兹2 sin共x 45 兲 59. y 苷兹2 sin x 61. y 苷 sin x 63. y 苷 20 sin x 4 6 3 3p y 苷 5兹2 sin x 4

1. sin 3x sin x 17. 29. 55. 65.

3.

1 共sin 8x sin 4x兲 2

5. sin 8x sin 2x

7.

1 共cos 4x cos 6x兲 2

y

y

69. 8π 3

2

71. 7π 4

2 x

2π 3

1 4

11.

冉冊

冉冊

67.

9.

−

y

y

73.

x

π 4

冉冊

75. 10

y 12

−2π 3

x

5π 6

13.

冉冊

17π 6

2

兹2 4

4π 3

x

−6

π 6

13π 6

x

−10

77. a. p共t兲苷 sin共2p 1336t兲 sin共2p 770t兲 83. identity

b. p共t兲苷 2 sin共2106p t兲 sin共566p t兲

c. 1053 cycles per second

79. identity

81. identity

Prepare for This Section (3.5), page 255 PS1. A one-to-one function is a function for which each range value (y value) is paired with one and only one domain value (x value). PS2. If every horizontal line intersects the graph of a function at most once, then the function is a one-to-one function. PS3. f 关g共x兲兴苷 x PS4. f 关 f 1共x兲兴苷 x PS5. The graph of f 1 is the reflection of the graph of f across the line given by y 苷 x. PS6. No

Exercise Set 3.5, page 265 1.

p 2

3.

5p 6

23. u 苷 cos1 47.

4兹15 15

67. cos

5.

冉冊 x 7

49.

24 25

5p ⬇ 0.2588 12

p 4

25.

7. 1 2

9.

27. 2

51. 0 69.

p 3

53.

p 3

29. 24 25

兹1 x 2 x

11.

p 4

3 5

31. 1

55.

2 兹15 6

13. 33. 57.

1 2

p 3 35.

15. p 6

2p 3 37.

1 共 3兹7 4兹3 兲 5

17.

p 6

p 4 59.

19. a. 1.0014

b. 0.2341

39. not defined

41. 0.4636

12 13

2 兹2 2

61. 2

63.

21. a. 1.1102 43. 65.

7兹2 10

p 6

b. 0.2818 45.

兹3 3

A18

Answers to Selected Exercises y

75.

y

77.

π

y

79.

81.

−4

y

4

83. a. s 苷 3960 cos1

x

x

−2

冉

冊

3960 a 3960

b. 17,930 miles

π

1

−π

x

1

y

85.

3

f g −1

f and g have the same graph in Quadrant I x 1

−π

x

π 2

87.

1 tan 5x 3

冉冊

99. y 苷 3 cos x

5 兹73 6

PS2. 1 cos2 x

9.5

0

− 2π

π − 2

p 3

Prepare for This Section (3.6), page 268 PS1. x 苷

12.5

2.35

−9.5

f苷g

2π

91.

− 12.5

− 2.35

−2

97. y 苷

π

89.

PS3.

5 9 13 p, p, and p 2 2 2

冉

PS4. 共x 1兲 x

兹3 2

冊

PS5.

100

0

PS6. 0, 1

40

0

Exercise Set 3.6, page 278 p 5p 7p 11p p 7p p 4p p p 3p 3p p 3p p p 3p 11p p 3p p p 5p , 3. , 5. , , , 7. , 9. , , , 11. , 13. , , 15. , , , 4 4 3 3 4 2 4 2 2 2 6 4 4 6 4 4 6 2 6 6 6 6 6 p 3p 5p 7p p 5p 4p 5p p 3p , , p, , , , , , p, 17. 0, 19. 21. 0, 23. 41.4 , 318.6 25. no solution 27. 68.0 , 292.0 29. no solution 4 4 4 4 6 6 3 3 2 2 31. 12.8 , 167.2 33. 15.5 , 164.5 35. 0 , 33.7 , 180 , 213.7 37. no solution 39. no solution 41. 0 , 120 , 240 43. 70.5 , 289.5 45. 68.2 , 116.6 , 248.2 , 296.6 47. 19.5 , 90 , 160.5 , 270 49. 60 , 90 , 300 51. 53.1 , 180 53. 72.4 , 220.2 55. 50.1 , 129.9 , 205.7 , p kp p 2kp 61. 334.3 57. no solution 59. 22.5 , 157.5 , where k is an integer 63. , where k is an integer 65. 0 2kp, 8 2 10 5 p 5p p 5p 2kp, p 2kp, 2kp, where k is an integer 67. kp, kp, where k is an integer 69. 0 2kp, where k is an integer 3 3 2 6 p p 5p 7p 3p 11p p 3p p 2p 4p 5p 4p 5p p 3p 5p 7p 71. 0, p 73. 0, , , , p, , , 75. 0, , 77. 0, , , p, , 79. , 81. 0, , , p, , 6 2 6 6 2 6 2 2 3 3 3 3 3 3 4 4 4 4 p 5p 83. , ,p 85. 0.7391 87. 3.2957, 3.2957 89. 1.16 91. 14.99 and 75.01 6 6 The sine regression functions in Exercises 93, 95, and 97 were obtained on a TI-83/TI-83 Plus/TI-84 Plus calculator by using an iteration factor of 16. The use of a different iteration factor may produce a sine regression function that varies from the regression functions listed below. 93. a. y ⬇ 1.1213 sin共0.01595x 1.8362兲 6.6257 b. 6⬊49 95. a. y ⬇ 49.2125 sin共0.2130x 1.4576兲 48.0550 b. 3% 97. a. y ⬇ 35.185 sin共0.30395x 2.1630兲 2.1515 b. 24.8 99. b. 42 and 79 c. 60 101. 0.93 foot, 1.39 feet p p 5p p p 3p 5p 3p 7p 103. , 105. ,0 107. 0, , , , p, , , 6 2 3 4 2 4 4 2 4 1.

Chapter 3 Assessing Concepts, page 286 1. True

2. False

3. False

4. True

5. 4

6.

1 1 x 2 2

7. 0 y p

8.

p 3

9. 兹2

10. 2 兹3

A19

Answers to Selected Exercises

Chapter 3 Review Exercises, page 286 1.

兹6 兹2 [3.2] 4

2. 兹3 2 [3.2]

3.

兹6 兹2 [3.2] 4

4. 兹2 兹6 [3.2]

5.

兹6 兹2 [3.2] 4

6.

兹2 兹6 [3.2] 4

7.

兹2 兹2 2

兹2 兹3

[3.3]

1 1 兹2 兹2 [3.3] 9. 兹2 1 [3.3] 10. [3.3] 11. a. 0 b. 兹3 c. [3.2/3.3] 12. a. 0 b. 2 c. [3.2/3.3] 2 2 2 2 兹3 兹2 兹3 兹6 兹2 13. a. b. 兹3 c. [3.2/3.3] 14. a. b. 兹3 c. 1 [3.2/3.3] 15. sin 6x [3.3] 16. tan 3x [3.2] 2 2 4 17. sin 3x [3.2] 18. cos 4u [3.3] 19. tan 2u [3.1] 20. tan u [3.3] 21. 2 sin 3u sin u [3.4] 22. 2 cos 4u sin u [3.4] 13 4 56 7 3 4 23. 2 sin 4u cos 2u [3.4] 24. 2 cos 3u sin 2u [3.4] 43. [3.5] 44. [3.5] 45. [3.5] 46. [3.5] 47. [3.6] 48. [3.6] 5 5 65 25 2 5 p p 49. 30 , 150 , 240 , 300 [3.6] 50. 0 , 45 , 135 [3.6] 51. 2kp, 3.8713 2kp, 5.553 2kp, where k is an integer. [3.6] 52. kp, 2 4 p 5p 13p 17p 7p 19p 3p 7p [3.6] 54. [3.6] 1.2490 kp, where k is an integer. [3.6] 53. , , , , , , 12 12 12 12 12 12 4 4 p 5p 4p p 55. y 苷 2 sin x [3.4] 56. y 苷 2兹2 sin x [3.4] 57. y 苷 2 sin x [3.4] 58. y 苷 sin x [3.4] 6 4 3 6 8.

冉冊

冉冊

y

π 6

−5π 4

11π x 6

冉冊

y 1

y

y 3

2 −

冉冊 2

3π 4

x

−

2π 3

4π 3

x

−1

−3

59.

[3.5]

y

60.

[3.5]

y

61.

π

[3.5]

y

π 2

62.

x

1

1

1

63. a. y ⬇ 1.1835 sin共0.01600x 1.8497兲 6.4394

x

π 2

x

x

1

13π 6

[3.5]

y

π 2

π 6

x

b. 6⬊01 [3.6]

Chapter 3 Quantitative Reasoning Exercises, page 288 QR1. a.

a ⬇ 0.87266

25

QR2. a.

1.35

0.1745

Maximum X=.8726644 −8

a ⬇ 0.87266

35

0.1745

Y=20.557941

Minimum X=.87266268 Y=23.917394

−8

Xscl = 0.2, Yscl = 10

Xscl = 0.2, Yscl = 10

Chapter 3 Test, page 289 兹6 兹2 [3.2] 4 5 15. [3.5] 16. 13 5.

6.

兹2 [3.2] 10

8. sin 9x [3.2]

[3.5]

y

y = sin−1 ( x + 2) −3

−1

9.

π 2

7 [3.3] 25

12.

2 兹3 [3.4] 4

17. 41.8 , 138.2 [3.6]

y = sin−1 x 1

2

3

x

π − 2

20. a. y ⬇ 1.7569 sin共0.01675x 1.3056兲 12.1100

1.35

b. 12 hours 1 minute [3.6]

18. 0,

冉冊

13. y 苷 sin x p 11p , p, [3.6] 6 6

5p [3.4] 14. 0.701 [3.5] 6 p 2p 4p 19. , , [3.6] 2 3 3

A20

Answers to Selected Exercises

Cumulative Review Exercises, page 290 1. x 3 [1.1]

2. Shift the graph of y 苷 f 共x兲 horizontally 1 unit to the left and up 2 units. [1.4] 3. Reflect the graph of y 苷 f 共x兲 across the x 4 2兹3 2兹5 兹3 [1.6] 6. [2.1] 7. 300° [2.1] 8. [2.2] 9. [2.2] 10. [2.2] x-axis. [1.4] 4. odd function [1.4/2.5] 5. f 1共x兲苷 x5 3 2 3 5 1 兹3 11. positive [2.3] 12. 50° [2.3] 13. [2.3] 14. x 苷 , y 苷 [2.4] 15. 0.43, , [2.7] 16. [3.5] 17. 2.498 [3.5] 3 2 2 12 6 7 11 18. 关1, 1兴 [3.5] 19. , [3.5] 20. , , [3.6] 2 2 2 6 6

冉

冊

Exercise Set 4.1, page 298 1. C 苷 77 , b ⬇ 16, c ⬇ 17 3. B 苷 38 , a ⬇ 18, c ⬇ 10 5. C ⬇ 15 , B ⬇ 33 , c ⬇ 7.8 7. C 苷 45.1 , b ⬇ 39.4, c ⬇ 30.2 9. C 苷 32.6 , c ⬇ 21.6, a ⬇ 39.8 11. B 苷 47.7 , a ⬇ 57.4, b ⬇ 76.3 13. A ⬇ 58.5 , B ⬇ 7.3 , a ⬇ 81.5 15. C 苷 59 , B 苷 84 , b ⬇ 46 or C 苷 121 , B 苷 22 , b ⬇ 17 17. No triangle is formed. 19. No triangle is formed. 21. C 苷 19.8 , B 苷 145.4 , b ⬇ 10.7 or C 苷 160.2 , B 苷 5.0 , b ⬇ 1.64 23. No triangle is formed. 25. C 苷 51.21 , A 苷 11.47 , c ⬇ 59.00 27. B ⬇ 130.9 , C ⬇ 28.6 , b ⬇ 22.2 or B ⬇ 8.1 , C ⬇ 151.4 , b ⬇ 4.17 29. ⬇ 68.8 miles 31. 231 yards 33. ⬇ 110 feet 35. 4840 feet 37. ⬇ 96 feet 39. ⬇ 33 feet 41. ⬇ 8.1 miles 43. ⬇ 1200 miles 45. ⬇ 260 meters 49. 48 minimum value of L ⬇ 11.19 meters

0

Minimum X=49.854625 Y=11.1941

90

− 15

Prepare for This Section (4.2), page 302 PS1. 20.7

PS2. 25.5 square inches

PS3. C 苷 cos1

冉

a 2 b2 c 2 2ab

冊

PS4. 12.5 meters

PS5. 6

PS6. c 2 苷 a2 b2

Exercise Set 4.2, page 308 1. ⬇ 13 3. ⬇ 150 5. ⬇ 29 7. ⬇ 9.5 9. ⬇ 10 11. ⬇ 40.1 13. ⬇ 90.7 15. ⬇ 39 17. ⬇ 90 19. ⬇ 47.9 21. ⬇ 116.67 23. ⬇ 80.3 25. a ⬇ 11.1, B ⬇ 62.0 , C ⬇ 78.6 27. A ⬇ 34.2 , B ⬇ 104.6 , C ⬇ 41.3 29. ⬇ 140 square units 31. ⬇ 53 square units 33. ⬇ 81 square units 35. ⬇ 299 square units 37. ⬇ 36 square units 39. ⬇ 7.3 square units 41. ⬇ 710 miles 43. ⬇ 74 feet 45. ⬇ 60.9 47. ⬇ 350 miles 49. 40 centimeters 51. ⬇ 2800 feet 53. 402 miles, S62 E 55. ⬇ 47,500 square meters 57. 162 square inches 59. ⬇ $41,000 61. ⬇ 6.23 acres 63. Triangle DEF has an incorrect dimension. 65. ⬇ 12.5 69. ⬇ 140 cubic inches

Prepare for This Section (4.3), page 312 PS1. 1

PS2. 6.691

PS3. 30

PS4. 157.6

PS5.

兹5 5

PS6.

14兹17 17

Exercise Set 4.3, page 325 1. a 苷 4, b 苷 2; 具4, 2典

冓

冔

3. a 苷 5, b 苷 4; 具5, 4典 5. a 苷 7, b 苷 1; 具7, 1典

冓冔

冔

7. a 苷 7, b 苷 5; 具7, 5典

冓

冔

9. a 苷 0, b 苷 8; 具0, 8典

冓

11. 5,

冔

7兹58 3兹58 兹5 2兹5 兹5 2兹5 13. ⬇ 44.7, ⬇ 296.6 , , 15. ⬇ 4.5, ⬇ 296.6 , , 17. ⬇ 45.7, ⬇ 336.8 , , 5 5 5 5 58 58 11 7 11 1 19. 具6, 12典 21. 具1, 10典 23. , 25. 2兹5 27. 2兹109 29. 8i 12j 31. 14i 6j 33. i j 35. 兹113 6 3 12 2 37. a1 ⬇ 4.5, a2 ⬇ 2.3, 4.5i 2.3j 39. a1 ⬇ 2.8, a2 ⬇ 2.8, 2.8i 2.8j 41. ⬇ 380 miles per hour 43. ⬇ 250 miles per hour at a heading of 86 45. 293 pounds 47. a. 131 pounds b. 319 pounds 49. The forces are in equilibrium. 51. The forces are not in equilibrium. F4 苷 0i 10j 53. The forces are in equilibrium. 55. 3 57. 0 59. 1 61. 0 63. ⬇ 79.7 65. 45 67. 90 , orthogonal 3 4 ⬇ 126.9 , , 5 5

69. 180

71.

46 5

冓

73.

14兹29 ⬇ 2.6 29

75. 兹5 ⬇ 2.2

77.

11兹5 ⬇ 4.9 5

79. ⬇ 954 foot-pounds

81. ⬇ 779 foot-pounds

Answers to Selected Exercises 83.

具6, 9典

y

A21

85. the vector from P1共3, 1兲 to P2共5, 4兲

(6, 9)

w

v

u+v

u+

v+

w

(1, 4)

(−1, 1) u x

87. Because v w 苷 0, the vectors are perpendicular.

89. 具7, 2典 is one example.

91. No

95. The same amount of work is done.

Chapter 4 Assessing Concepts, page 329 1. a triangle that does not contain a right angle 2. the Law of Cosines 6. a scalar 7. True 8. False 9. True 10. True

3. SSA

4. the semiperimeter of a triangle

5. a scalar

Chapter 4 Review Exercises, page 329 1. B 苷 51 , a ⬇ 11, c ⬇ 18 [4.1] 2. A 苷 8.6 , a ⬇ 1.77, b ⬇ 11.5 [4.1] 3. B ⬇ 48 , C ⬇ 95 , A ⬇ 37 [4.2] 4. A ⬇ 47 , B ⬇ 76 , C ⬇ 58 [4.2] 5. c ⬇ 13, A ⬇ 55 , B ⬇ 90 [4.2] 6. a ⬇ 169, B ⬇ 37 , C ⬇ 61 [4.2] 7. No triangle is formed. [4.1] 8. No triangle is formed. [4.1] 9. C 苷 45 , a ⬇ 29, b ⬇ 35 [4.1] 10. A 苷 115 , a ⬇ 56, b ⬇ 26 [4.1] 11. ⬇ 360 square units [4.2] 12. ⬇ 31 square units [4.2] 13. ⬇ 920 square units [4.2] 14. ⬇ 46 square units [4.2] 15. ⬇ 790 square units [4.2] 16. ⬇ 210 square units [4.2] 17. ⬇ 170 square units [4.2] 18. ⬇ 140 square units [4.2] 19. a1 苷 5, a2 苷 3, 具5, 3典 [4.3] 20. a1 苷 1, a2 苷 6, 具1, 6典 [4.3] 8兹89 5兹89 21. ⬇ 4.5, 153.4 [4.3] 22. ⬇ 6.7, 333.4 [4.3] 23. ⬇ 3.6, 123.7 [4.3] 24. ⬇ 8.1, 240.3 [4.3] 25. , [4.3] 89 89 7兹193 12兹193 5兹26 3兹34 5兹34 兹26 26. , [4.3] 27. i j [4.3] 28. i j [4.3] 29. 具7, 3典 [4.3] 30. 具18, 7典 [4.3] 193 193 26 26 34 34 17 13 47 31. 6i j [4.3] 32. i j [4.3] 33. 420 mph [4.3] 34. ⬇ 7 [4.3] 35. 18 [4.3] 36. 21 [4.3] 37. 9 [4.3] 2 6 6 27兹29 10兹41 38. 20 [4.3] 39. ⬇ 86 [4.3] 40. ⬇ 138 [4.3] 41. ⬇ 125 [4.3] 42. ⬇ 157 [4.3] 43. [4.3] 44. [4.3] 41 29

冓

冓

冔

冔

45. ⬇ 662 foot-pounds [4.3]

Chapter 4 Quantitative Reasoning Exercises, page 330 QR1. 2210 miles QR2. 289 QR3. 1620 miles QR4. 82

Chapter 4 Test, page 332 1. B 苷 94 , a ⬇ 48, b ⬇ 51 [4.1] 2. ⬇ 11 [4.1] 3. ⬇ 14 [4.2] 4. ⬇ 48 [4.2] 5. ⬇ 39 square units [4.2] 6. ⬇ 93 square units [4.2] 7. ⬇ 260 square units [4.2] 8. 兹13 [4.3] 9. 9.193i 7.713j [4.3] 10. 19i 29j [4.3] 11. 1 [4.3] 12. 103 [4.3] 13. ⬇ 27 miles [4.2] 14. ⬇ 21 miles [4.1] 15. ⬇ $65,800 [4.2]

Cumulative Review Exercises, page 332 1. 兹74 units [1.2]

2. sin x cos x [1.5]

6. 33 centimeters [2.2]

7.

[2.5]

y 4

8.

−4

[2.6]

y

5. shifted 2 units to the right and 3 units up [1.4] y 4

9.

4

2 −4 −2 −2

4. f 1(x) 苷 2x 6 [1.6]

3. sec(cos x) [1.5]

2

2

2

4 x

−π

−2 −4

[2.7]

π

2π

x

−2

−2 −4

2 x

A22

Answers to Selected Exercises

10. amplitude: 3; period: 6p, phase shift:

3p [2.7] 2

11. amplitude: 兹2, period: 2p, phase shift:

p 5 p 5p [3.5] 15. [3.5] 16. 0, , , p [3.6] 3 12 6 6 20. ground speed: 592 mph, heading: 67.4° [4.3] 13. See [3.1]

p [2.7] 4

12. 9.5 centimeters [4.2]

17. magnitude: 5; angle: 323.1 [4.3]

14.

18. 60.3 [4.3]

19. 289.5 [4.3]

Exercise Set 5.1, page 340 1. 9i

5. 4 9i

3. 7i兹2

23. 10

27. 20 10i

25. 19i

43. 1 i

45.

7. 5 7i

15 29 i 41 41

47.

9. 8 3i兹2

29. 22 29i

12 5 i 13 13

11. 11 5i

31. 41

49. 2 5i

13. 7 4i

33. 12 5i

51. 16 30i

15. 8 5i

35. 114 42i兹2 53. 11 2i

37. 6i

55. i

63. z1 苷 2 4i, z2 苷 6 7i, z3 苷 10 10i, 65. z1 苷 1 i, z2 苷 1 i, z3 苷 1 i, z4 苷 1 i, z4 苷 14 13i, z5 苷 18 16i z5 苷 z1, z6 苷 z2, z7 苷 z3, z8 苷 z4 1 1 i 71. 73. 共x 4i 兲共x 4i 兲 75. 共z 5i 兲共z 5i 兲 77. 共2x 9i 兲共2x 9i 兲 83. 0 2 2

17. 10

21. 40 7 2 41. i 53 53

39. 3 6i

57. 1 67.

19. 2 16i

59. i

1 兹3 i 2 2

69.

61. 1 3 兹3 i 2 2

Prepare for This Section (5.2), page 343 PS1. 1 3i

PS2.

1 1 i 2 2

PS3. 2 3i

PS4. 3 5i

PS5.

1 兹3

i 2 2

PS6. 3i, 3i

Exercise Set 5.2, page 349 1–7.

9. 兹2 cis 315

Im Re

−2 − 2i

−2i

21. 兹2 i兹2

3−i

35. 3 − 5i

兩2 2i兩苷 2兹2 兩兹3 i兩苷 2 兩2i兩苷 2 兩3 5i兩苷兹34

9兹3 9 i 2 2

47. 4 4i兹3 59.

1 兹3 i 2 2

11. 2 cis 330 13. 3 cis 90 15. 5 cis 180 17. 16 cis 120 兹2 兹2 23. 25. 3兹2 3i兹2 27. 8 29. 兹3 i i 2 2 37. ⬇ 0.832 1.819i 49. 3i

51.

39. 6 cis 255

3兹3 3i 2 2

61. 0 兹2 i 苷兹2 i

41. 12 cis 335

53. ⬇ 2.081 4.546i 63. 16 16i

65.

19. 4 cis 240 31. 3i

43. 10 cis

16 15

55. ⬇ 2.732 0.732i

3 兹3 i 8 8

67. 59.0 43.0i

33. 4兹2 4i兹2 45. 24 cis 6.5 57. 6 0i 苷 6 69. r 2 or a2 b2

Prepare for This Section (5.3), page 350 PS1. i

PS2. 2

PS3. 3

PS4. 2兹2 cis

4

PS5. 兹3 i

PS6. 1

Exercise Set 5.3, page 354 1. 128 128i兹3 3. 16 16i兹3 5. 16兹2 16i兹2 7. 64 0i 苷 64 9. 0 32i 苷 32i 11. 4 0i 苷 4 13. 1024 1024i 15. 0 1i 苷 i 17. 3 0i 苷 3 19. 2 0i 苷 2 21. 0.809 0.588i 23. 1 0i 苷 1 25. 1.070 0.213i 27. 0.276 1.563i 3 0i 苷 3 1 i兹3 0.309 0.951i 1 i兹3 0.213 1.070i 1.216 1.020i 1 i兹3 1 0i 苷 1 1.070 0.213i 1.492 0.543i 2 2 2 0i 苷 2 0.309 0.951i 0.213 1.070i 1 i兹3 1 i兹3 0.809 0.588i 2 2 1 i兹3 4 3 29. 2兹2 2i兹6 31. 2 cis 60 33. cis 67.5 35. 3 cis 0 37. 3 cis 45 39. 兹 2 cis 75 41. 兹 2 cis 80 4 3 2兹2 2i兹6 2 cis 180 cis 157.5 3 cis 120 3 cis 135 2 cis 165 2 cis 200 兹兹 4 3 2 cis 300 cis 247.5 3 cis 240 3 cis 225 2 cis 255 2 cis 320 兹兹 4 cis 337.5 3 cis 315 2 cis 345 兹 47. For n 2, the sum of the nth roots of 1 is 0.

A23

Answers to Selected Exercises

Chapter 5 Assessing Concepts, page 358 1. True 2. True 3. True 8. No 9. 3 5i 10. 2

4. False

5. The four roots are equally spaced around a circle with center (0, 0) and radius 1.

6. 1

7. 5

Chapter 5 Review Exercises, page 358 1. 3 8i, 3 8i [5.1] 7. 5 2i [5.1]

2. 6 2i, 6 2i [5.1]

8. 3 4i [5.1]

9. 3 3i [5.1]

26 15 i [5.1] 15. 2 2i [5.1] 53 53 21. i [5.1] 22. i [5.1] 23. 1 [5.1] 14.

30. 35. 40. 45. 50. 54. 57. 59.

3. 2 i兹5, 2 i兹5 [5.1] 10. 5 15i [5.1]

16. 2 11i [5.1] 24. 1 [5.1]

17. 6 6i [5.1]

4. 5 3i兹3, 5 3i兹3 [5.1]

11. 25 19i [5.1]

5. 4 [5.1]

12. 29 2i [5.1]

18. 5 6i [5.1]

6. 9 [5.1]

8 6 13. i [5.1] 25 25

20. 11 i兹5 [5.1]

19. 7 [5.1]

27. 兹41 [5.1] 28. 兹2 [5.1] 29. 2兹2 cis 315° [5.2] 5兹2 5兹2 2 cis 150° [5.2] 31. ⬇ 兹13 cis 146.3° [5.2] 32. 兹17 cis 345.96° [5.2] 33. i ⬇ 3.536 3.536i [5.2] 34. 3 3i兹3 [5.2] 2 2 ⬇ 0.832 1.819i [5.2] 36. 1.27 2.72i [5.2] 37. 0 30i 苷 30i [5.2] 38. 5兹2 5i兹2 [5.2] 39. ⬇ 8.918 8.030i [5.2] ⬇ 0.968 3.881i [5.2] 41. ⬇ 6.012 13.742i [5.2] 42. ⬇ 13.2 27.0i [5.2] 43. 3 cis 共100°兲 [5.2] 44. 3 cis 110° [5.2] 5 1 243兹2 243兹2 i ⬇ 171.827 171.827i [5.3] 5 cis 共59°兲 [5.2] 46. 兹2 cis 共50°兲 [5.2] 47. cis 1.9 [5.2] 48. cis 共4兲 [5.2] 49. 3 2 2 2 1 [5.3] 51. 64 64兹3i ⬇ 64 110.851i [5.3] 52. 32,768i [5.3] 53. 16兹2 16i兹2 ⬇ 22.627 22.627i [5.3] 4 4 4 4 237 3116i [5.3] 55. 3 cis 30°, 3 cis 150°, 3 cis 270° [5.3] 56. 兹 8 cis 22.5°, 兹 8 cis 112.5°, 兹 8 cis 202.5°, 兹 8 cis 292.5° [5.3] 4 cis 0 , 4 cis 90 , 4 cis 180 , 4 cis 270 [5.3] 58. 2 cis 45°, 2 cis 117°, 2 cis 189°, 2 cis 261°, 2 cis 333° [5.3] 3 cis 0°, 3 cis 90°, 3 cis 180°, 3 cis 270° [5.3] 60. 5 cis 60°, 5 cis 180°, 5 cis 300° [5.3] 25. 8 [5.1]

26. 兹13 [5.1]

Chapter 5 Quantitative Reasoning Exercises, page 359 QR1.

QR2. 0.25 0.25i, 1 0.1i, and 0.1 0.2i QR3. 2 is an element of the Mandelbrot set because all of its iterates equal 2. QR4. The first iterate of 2i is z1 苷 4 2i. 2i is not an element of the Mandelbrot set because 兩4 2i兩 2. QR5. 共Z2 S兲 l Z QR6. Answers will vary.

2 Im-axis

−2

1.03 0

Chapter 5 Test, page 360 3. 10 i [5.1] 4. 9 9i [5.1] 5. 6 [5.1] 6. i [5.1] 7. 5 16i [5.1] 8. 3 54i [5.1] 13 34 9. 16 30i [5.1] 10. 5 4i [5.1] 11. i [5.1] 12. 2 4i [5.1] 13. 兹34 [5.1] 14. 3兹2 cis 315° [5.2] 15. 6 cis 270° [5.2] 25 25 5兹2 5兹2i 16. 2 2i兹3 [5.2] 17. [5.2] 18. 6兹2 6i兹2 [5.2] 19. ⬇ 11.472 16.383i [5.2] 20. 4 [5.2] 21. 6i [5.2] 2 2 3 3 3 22. 16,777,216 [5.3] 23. 2 cis 0°, 2 cis 60°, 2 cis 120°, 2 cis 180°, 2 cis 240°, 2 cis 300° [5.3] 24. 兹 2 cis 40°, 兹 2 cis 160°, 兹 2 cis 280° [5.3] 25. 2 cis 36°, 2 cis 108°, 2 cis 180°, 2 cis 252°, 2 cis 324° [5.3] 1. 6 3i [5.1]

2. 3i兹2 [5.1]

Cumulative Review Exercises, page 361 1. 关2, 3兴 [1.1]

2. All real numbers except 2 and 2 [1.3]

7. ⬇25 centimeters [2.2]

8. a 苷 1, b 苷 1 [2.4]

3. 2 [1.3]

4. 共f ⴰ g兲共x兲苷 sin共x 2 1兲 [1.5]

y

9.

[2.5]

y

10.

4

−4 −2 −4

11. See [3.1]

12. sin x [3.2]

18. 1449 foot-pounds [4.3]

13.

56 [3.2] 65

14.

19. 2兹2 cis 45° [5.2]

5.

3 [1.6] 2

6. 270° [2.1]

[2.6]

4

2 4

x

−4

4

x

−4

p 11p 15. 0, , p, [3.6] 16. ⬇ 170 centimeters [4.2] 17. 64.7° [4.3] 6 6 3 3兹3 3 3兹3 20. 3, i, i [5.3] 2 2 2 2

33 [3.5] 65

A24

Answers to Selected Exercises

Exercise Set 6.1, page 370 1. a. iii b. i 3. vertex: 共0, 0兲

c. iv d. ii 5. vertex: 共0, 0兲 1 focus: 共0, 1兲 focus: ,0 12

7. vertex: 共2, 3兲

冉冊

1 directrix: x 苷 12

directrix: y 苷 1

2 x

4

directrix: y 苷 5

directrix: x 苷 3

y

directrix: x 苷

2

x

2

−4

x

2

2 −4

−4

13. vertex: 共2, 2兲

15. vertex: 共4, 10兲

冉冊

冉

5 2

focus: 2,

冊

冉冊冉冊

7 3 , 4 2 3 focus: 2, 2

17. vertex:

39 4 41 directrix: y 苷 4

focus: 4,

3 directrix: y 苷 2 y

y 8 4

2

−4

冉冊冉冊

5 4 3 focus: 2, 4

23. vertex: 2,

directrix: y 苷

7 4

−2

y

2

x

冉冉

冊冊

3 13 , 2 12 3 1 focus: , 2 3 11 directrix: y 苷 6

21. vertex:

y 2 x

4

x

x

冉冊冉冊

1 9 31 focus: 1, 36

29. x 2 苷 16y

27. vertex: 1,

directrix: y 苷

31. 共x 1兲2 苷 4共 y 2兲

23 36

y

y 4

y 8

y

3

−4

冉冊冉冊

冊

2

9 , 3 2 11 directrix: x 苷 2

2

9 , 1 2 35 focus: , 1 8 37 directrix: x 苷 8

25. vertex:

冉

focus:

x

x

2

19. vertex: 共5, 3兲

3 directrix: x 苷 2

9 2

y

y 2

x

−2

冉冊

focus: 共1, 4兲

2

−4

11. vertex: 共4, 1兲 7 focus: , 1 2

focus: 共2, 1兲

y

y

9. vertex: 共2, 4兲

4

x

2

x

4

−2

33. 共x 3兲2 苷 4共 y 4兲 above vertex

冉冊

35. 共x 4兲2 苷 4共 y 1兲

41. 6.0 inches

2

4

x

37. vertex: 共250, 20兲, focus:

43. a. 5900 square feet

b. 56,800 square feet

冉

3240 , 20 13

冊

39. on axis of symmetry 4 feet

45. a 苷 1.5 inches

2

47. a. 4

800 共y 32兲苷 x 2 b. 117 millimeters 49. 4 84

51. 4兩p兩

53.

y

55.

y

2 2

x

2 2

x

Answers to Selected Exercises 57. x 2 y 2 8x 8y 2xy 苷 0

Prepare for This Section (6.2), page 374 PS1. midpoint: 共2, 3兲; length: 2兹13 PS6. y −2

PS3. 1 兹3

PS4. x 2 8x 16 苷共x 4兲2

PS5. y 苷兹4 共x 2兲2

6x

4

2

−2

PS2. 8, 2

−4 −6

Exercise Set 6.2, page 385 1. a. iv b. i c. ii d. iii 3. center: 共0, 0兲 5. center: 共0, 0兲 vertices: 共0, 5兲, 共0, 5兲 vertices: 共3, 0兲, 共3, 0兲

7. center: 共0, 0兲 vertices: 共3, 0兲, 共3, 0兲

foci: 共兹5, 0 兲, 共兹5, 0 兲

foci: 共兹2, 0 兲 , 共兹2, 0 兲

foci: 共0, 3兲, 共0, 3兲 y 6 4

y

y

y

4

4

x −4

−3

13. center: 共2, 0兲 vertices: 共2, 5兲, 共2, 5兲 foci: 共2, 4兲, 共2, 4兲

x

−6

−4

y

y

2

6 4 2

−2 −2

2 4

x −2

17. center: 共1, 1兲 vertices: 共1, 2兲, 共1, 4兲兹65 兹65 , 1, 1 foci: 1, 1 3 3

冊

21. center: 共0, 0兲 vertices: 共0, 5兲, 共0, 5兲

foci: 共1, 0兲, 共1, 0兲

foci: 共0, 3兲, 共0, 3兲 y 6

2 −2 −4

4

x

x

19. center: 共0, 0兲 vertices: 共2, 0兲, 共2, 0兲

y

y

2 4

1 1

x

4x

冊

4

15. center: 共1, 3兲 vertices: 共 1 兹21, 3 兲, 共 1 兹21, 3 兲 foci: 共 1 兹17, 3 兲, 共 1 兹17, 3 兲

−8

冊冉

−4

−4

y

冉

x

4

2 2 4 6

冉冊冉

3 x

11. center: 共3, 2兲 vertices: 共8, 2兲, 共2, 2兲 foci: 共6, 2兲, 共0, 2兲

9. center: 共0, 0兲 vertices: 共0, 4兲, 共0, 4兲兹55 兹55 foci: 0, , 0, 2 2

x

A25

A26

Answers to Selected Exercises

23. center: 共0, 0兲 vertices: 共0, 4兲, 共0, 4兲兹39 兹39 foci: 0, , 0, 2 2

冉冊冉

25. center: 共3, 4兲 vertices: 共3, 6兲, 共3, 2兲

冊

27. center: 共2, 3兲 vertices: 共1, 3兲, 共5, 3兲

foci: 共 3, 4 兹3 兲, 共 3, 4 兹3 兲

y

foci: 共0, 3兲, 共4, 3兲 y 2

y

2

4 x

1

−2

2

x

4 x

2

29. center: 共2, 0兲

6

31. center: 共1, 1兲

33. center: 共3, 1兲

冉冊冉冊冉冊冉

vertices: 共2, 4兲, 共2, 4兲

vertices: 共1, 6兲, 共1, 4兲

vertices:

11 , 1 , 2

foci: 共 2, 兹7 兲, 共 2, 兹7 兲

foci: 共1, 4兲, 共1, 2兲

foci: 3

兹17 兹17 , 1 , 3 , 1 2 2

y

y

y

1 , 1 2

冊

6

4

6

x

6

x

4 x

−2

x2 y2 x2 y2 x2 y2 共x 2兲2 共 y 4兲2 共x 2兲2 共 y 4兲2 苷1 苷1 苷1 苷1 苷1 37. 39. 41. 43. 25 9 36 16 36 81兾8 16 7 25兾24 25 2 2 2 2 2 2 2 2 2 2 共x 5兲共 y 1兲 x y x y 共x 1兲共 y 3兲 x y 苷1 苷1 苷1 苷1 苷1 45. 47. 49. 51. 53. 16 25 25 21 20 36 25 21 80 144 2 2 x y 苷1 55. on the major axis of the ellipse, 41 centimeters from the emitter 57. 59. 40 feet 884.742 883.352 2 9兹15 x 2 y2 苷1 61. 63. 24 feet 65. a. 兹7 feet to the right and left of O. b. 8 feet 324 81兾4 35.

冉

67. y 苷

冊

36 兹1296 36共16x 2 108兲 18

69. y 苷

18 兹324 36共4x 2 24x 44兲 18

6.2

−9.4

75.

2

−4

9.4

71.

x2 y2 苷1 36 27

73.

共x 1兲2 共 y 2兲2 苷1 16 12

9 2

0.7

−6.2

−2

Prepare for This Section (6.3), page 390 PS1. midpoint: 共1, 1兲; length: 2兹13

PS2. 4, 2

PS3. 兹2

PS4. 4共x 2 6x 9兲苷 4共x 3兲2

PS5. y 苷

3 兹x 2 4 2

Answers to Selected Exercises PS6.

y

−2

2

−2

4

6x

−4 −6

Exercise Set 6.3 page 399 1. a. iii b. ii 3. center: 共0, 0兲

c. i

d. iv 5. center: 共0, 0兲

7. center: 共0, 0兲

9. center: 共0, 0兲

vertices: 共 4, 0兲

vertices: 共0, 2兲

vertices: 共兹7, 0 兲

foci: 共兹41, 0 兲

foci: 共 0, 兹29 兲

foci: 共 4, 0兲

asymptotes: y 苷

5 x 4

asymptotes: y 苷

2 x 5

3 ,0 2 兹73 ,0 foci:

2

asymptotes: y 苷

y

y

冉冊冉冊

vertices:

3兹7 x 7

asymptotes: y 苷

y 4

y 4

4

8 x 3

2 x

2

x

6

4 x

−4

−4

−4

11. center: 共3, 4兲 vertices: 共7, 4兲, 共1, 4兲 foci: 共8, 4兲, 共2, 4兲 3 asymptotes: y 4 苷共x 3兲 4

13. center: 共1, 2兲 vertices: 共1, 0兲, 共1, 4兲 foci: 共 1, 2 2兹5 兲 asymptotes: y 2 苷

y

y

4

2

15. center: 共2, 0兲 vertices: 共1, 0兲, 共5, 0兲 foci: 共 2 兹34, 0 兲 5 asymptotes: y 苷共x 2兲 3

1 共x 1兲 2

y

x

4

4 x

−4

6

2

x

2 x

−4 −4

17. center: 共1, 1兲 7 1 , 1 , , 1 vertices: 3 3 兹97 , 1 foci: 1

3 9 asymptotes: y 1 苷共x 1兲 4

冉冊冉冉冊 4

2 −4

4

x

21. center: 共0, 0兲

23. center: 共0, 0兲

冉冊冉冊

vertices: 共 3, 0兲

vertices: 共0, 3兲

vertices: 0,

foci: 共 3兹2, 0 兲

foci: 共0, 5兲

foci: 0,

asymptotes: y 苷 x

asymptotes: y 苷

y 6

y

−4

冊

19. center: 共0, 0兲

兹5 3

asymptotes: y 苷 2x y

y 6

6 x

3 x 4

6 x

2 3

1

x

A27

A28

Answers to Selected Exercises

25. center: 共3, 4兲 vertices: 共3, 6兲, 共3, 2兲 foci: 共 3, 4 2兹2 兲 asymptotes: y 4 苷共x 3兲

27. center: 共2, 1兲 vertices: 共2, 2兲, 共2, 4兲 foci: 共 2, 1 兹13 兲 3 asymptotes: y 1 苷共x 2兲 2

y

29. y 苷

14

y 6

4

6 兹36 4共4x 2 32x 39兲 2

−8.75

3

4 x

8

−5

x

64 兹4096 64共9x 2 36x 116兲 32

31. y 苷

33. y 苷

18 兹324 36共4x 2 8x 6兲 18

6

35.

x2 y2 苷1 9 7

37.

y2 x2 苷1 20 5

6

−3.75

−6.75

8

5

−8

−8

y2 x2 y2 x2 共x 4兲2 共 y 3兲2 共x 4兲2 共 y 2兲2 共 y 2兲2 共x 7兲2 苷1 苷1 苷1 苷1 苷1 41. 43. 45. 47. 9 36兾7 16 64 4 5 144兾41 225兾41 3 12 x2 共 y 7兲2 共x 1兲2 x2 y2 共x 4兲2 共 y 1兲2 共 y 1兲2 共x 4兲2 y2 苷1 苷1 苷 1 and 苷1 苷1 49. 51. 53. 55. a. 1 3 4 12 36兾7 4 36兾7 4 2162.25 13,462.75 2 2 y x 苷 1 b. 6.25 inches b. 221 miles 57. y 2 x 2 苷 10,000 2, hyperbola 59. a. 2 2 0.52 x2 y2 苷1 61. ellipse 63. parabola 65. parabola 67. ellipse 69. 1 3 39.

y

y

y 10

y

10

4

71.

y2 x2 苷1 9 7

10

2

x

4

−6

−2

x

x

6

x

−8

y

73.

2 −2

2

6

x

−6

Prepare for This Section (6.4), page 403 PS1. cos a cos b sin a sin b PS6.

y 4

2 −2

2 −4

4

x

PS2. sin a cos b cos a sin b

PS3.

p 6

PS4. 150°

PS5. hyperbola

A29

Answers to Selected Exercises

Exercise Set 6.4, page 410 1. 45°

3. 36.9°

5. 73.5°

7. 22.5°

9. 45°,

( x )2 ( y )2 苷1 8 8

11. 18.4°,

y 8

y'

x' x

4

4 −4

−4

17. 36.9°, ( y )2 苷 2共 x 2兲

y

4

2

21.

−4

6

x

4

−4

x' ≈26.6° x

−4 −8

x

4 −4

23.

25.

6.5

−9.4

−4.7

−4

8

4

x' 36.9°

4 4

−4

6

−8

−4

y'

x' 36.9°

4

4

4 x' 30° 4 x

4

y

y'

4

x' ≈18.4° x

19. 36.9°, 15( x )2 10( y )2 6x 28y 11 苷 0

y

y'

y

8

4

−8

15. 30°, y 苷 ( x )2 4

4

4

( x )2 ( y )2 苷1 1兾2 1兾3

y'

45° 4

13. 26.6°,

y

y'

4

8

( x )2 ( y )2 苷1 9 3

6.5

−9.4

9.4

9.4

4.7 0

−6.5

冉

3兹15 兹15 兹3 2兹2 兹3 2兹2 x and y 苷 x 29. , 5 5 2兹3 兹2 2兹3 兹2 2 2 37. hyperbola 39. ellipse 43. 9x 4xy 6y 苷 100 27. y 苷

冊冉

and

3兹15 兹15 , 5 5

−6.5

冊

31. hyperbola

33. parabola

35. parabola

Prepare for This Section (6.5), page 412 PS1. odd

PS2. even

PS3.

2p 5p , 3 3

PS5. r 2

PS4. 240°

PS6. 共4.2, 2.6兲

Exercise Set 6.5, page 423 1–7.

9.

5π

− 3, 3

11.

13.

15.

(2, 60°) π

−2, 4

(1, 315°)

4

4

21.

23.

0 u 2p 17.

19.

0up

0 u 2p

6

0up

5

2

6

6

0 u 2p

0 u 2p 4

25.

5

0 u 2p

−3

0 u 2p

9

−4

A30

Answers to Selected Exercises

27.

4

−6

0up

29.

−6

6

0 u 4p

35.

9

37.

−72

12

0 u 2p

41. 共2, 60 兲

0 u 2p

48

72

−8

10

−8

0 u 6p

8

−4

39.

−9

−6

−12

6

0up

6

−4

4

−6

31.

6

−4

33.

0up

4

−48

43.

冉

3 3兹3 , 2 2

冊

45. 共0, 0兲

47. 共5, 53.1 兲

49. x 2 y 2 3x 苷 0

8

−10

51. x 苷 3

53. x 2 y 2 苷 16

67. r 苷 3 sec u

69. r 苷 2

55. y 苷

兹3 x 3

57. x 4 y 2 x 2y 2 苷 0

71. r cos2 u 苷 8 sin u

61. y 苷 2x 6

59. y 2 4x 4 苷 0

73. r 2共cos 2u兲苷 25

3

75.

63. r 苷 2 csc u

77.

−4

4

79.

2

−2

2

81. −4

85.

87.

10

4 −3

−4

−2

2

3

−15

−4

−2

−10

0up 3

89.

15

4

−2

20

91.

−30 −3

30

0 u 2p

4

93. a.

−5

4

b.

5

−5

5

3 −1

−20

−4

4p u 4p

30 u 30

0 u 5p

p 3

2

−3 5

65. u 苷

−4

0 u 20p

A31

Answers to Selected Exercises Prepare for This Section (6.6), page 426 PS1.

3 5

PS2. x 苷 1

PS3. y 苷

2 1 2x

PS4.

3p 2

PS5. e 1

PS6.

4 2 cos x

Exercise Set 6.6, page 431 1. hyperbola

3. ellipse

5. parabola

8

冉冊冉冊

7. hyperbola

9. ellipse with holes at p 3p 2, and 2, 2 2

4

4

15 5

11. ellipse with holes at (2, 0) and (2, p)

15. 3x 2 y 2 16x 16 苷 0

13. parabola

5

5

19. x 2 6y 9 苷 0 31. r 苷

3 1 2 sin u

21. r 苷

2 1 2 cos u

23. r 苷

2 1 sin u

6.2

33.

−9.4

25. r 苷

8 3 2 sin u

27. r 苷

9.2

35.

6 2 3 cos u

29. r 苷

4 1 cos u

6.2

37.

−9.4

9.4 −9.4

9.4

9.4

− 6.2

−3.2

− 6.2

Rotate the graph in Exercise 1 p counterclockwise radian. 6

Rotate the graph in Exercise 3 counterclockwise p radians.

Rotate the graph in Exercise 5 p clockwise radian. 6

6.2

39.

17. 16x 2 7y 2 48y 64 苷 0

4

41.

− 14.4

4.4

−3

4

43.

4

4

−5

−5

5

5

−2

−4

−6.2

45.

Rotate the graph in Exercise 7 clockwise p radians.

−4

0 u 12p

Prepare for This Section (6.7), page 432 PS1. y 2 3y

冉冊

9 3 苷 y 4 2

2

PS2. y 苷 4t 2 4t 1

PS3. ellipse

PS4. 1

PS5. 共6, 2兲

PS6. domain: 共, 兲; range: 关3, 3兴

A32

Answers to Selected Exercises

Exercise Set 6.7, page 440 y

1.

y

3.

4

4 x

9.

4

1

y

4

4 x

23. The point traces a portion of the top branch of the hyperbola y2 x2 1, as shown in the figure. The point starts at (1, 兹2) and moves along the hyperbola until it reaches p the point (1, 兹2) at time t 苷 . 2 y

2

2 x 2 0 < t ≤ π/2

(−1, −1)

x

x

21. The point traces a line segment, as shown in the figure. The point starts at (1, 1) and moves along the line segment until it reaches the point (5, 4) at time t 3.

(5, 3)

−2

y

(2, 3)

2 2

19. The point traces the top half of the ellipse y 共x 2兲2 共 y 3兲2 苷 1, as shown in 32 22 4 the figure. The point starts at (5, 3) and (−1, 3) 2 moves counterclockwise along the ellipse until it reaches the point (1, 3) at time t .

27.

y

x

17. y 苷 x 2 1 x0 y 1

1

4 x

13. y 苷 2x 7 x2 y3

y

1 1

y

4

4 x

4 x

15. x 2/3 y 2/3 苷 1 1 x 1 1 y 1

y

7.

4

4 x

11. x 2 y 2 1 苷 0 x1 yR

y

y

5.

4

2

4

y (5, 4) 4 (−1, 1) 2

6 x

−2

(1, 2)

(−1, 2)

(0, 1) −1

x 2 π < t ≤ 3π/2

1

x

5

29.

500

31. a. x 苷 6, y 苷 60t, for t 0

−1

b. the Hummer

4.5π

35.

700

37.

5

−7

0

1500 0

Max height (nearest foot) of 462 feet is attained when t ⬇ 5.38 seconds. Range (nearest foot) of 1295 feet is attained when t ⬇ 10.75 seconds.

4

25. C1: y 苷 2x 5, x 2; C2: y 苷 2x 5, x R. C2 is a line. C1 is a ray.

y

−1

33.

2

0

7

3500 0

Max height (nearest foot) of 694 feet is attained when t ⬇ 6.59 seconds. Range (nearest foot) of 3084 feet is attained when t ⬇ 13.17 seconds.

−5

6 x

A33

Answers to Selected Exercises

冉冊冉冊

43. x 苷 a cos a sin

5

39.

45. x 苷共b a兲 cos a cos

ba a

y 苷共b a兲 sin a sin

ba a

y 苷 a sin a cos −5

5

−5

Chapter 6 Assessing Concepts, page 445 1. d

2. b

3. e

4. c

5. a

6. f

7. g

8. i

9. h

10. j

11. k

12. k

Chapter 6 Review Exercises, page 446 1. center: 共0, 0兲 vertices: 共 2, 0兲 foci: 共 2兹2, 0 兲 asymptotes: y 苷 x

4. center: (2, 3) vertices: 共0, 3兲, 共4, 3兲 foci: 共2 兹7, 3兲

[6.1]

y

[6.2]

y

8

4

y

−4

2

x

4

x

4

兹3 共x 2兲 2 [6.3]

asymptotes: y 3 苷

[6.3]

y

3. center: 共3, 1兲 vertices: 共1, 1兲, 共7, 1兲 foci: 共 3 2兹3, 1 兲

2. vertex: (0, 0) focus: (4, 0) directrix: x 苷 4

−8 − 4 −4

4

8 x

4

−3

x

−8

5. vertex: 共2, 1兲 29 focus: , 1 16

冉

冊

directrix: x 苷

7. center: 共2, 1兲

6. vertex: (3, 1) 21 ,1 focus: 8

冉冊

35 16

directrix: x 苷

27 8

8. center: 共2, 1兲

vertices: 共2, 2兲, 共2, 4兲

vertices: 共7, 1兲, 共3, 1兲

foci: 共 2, 1 兹5 兲

foci: 共8, 1兲, 共4, 1兲兹11 共x 2兲 5 [6.3]

asymptotes: y 1 苷

[6.1]

y

[6.1]

y

4

[6.2]

y

8

y 8

4

4

6 −8

x

−4

4

8 x

4

x

−8

冉冊冉冊冉冊冉冊 2 3

2 2 , 7, 3 3 2 foci: 1 2兹13, 3 2 2 asymptotes: y 苷共x 1兲 3 3 [6.3] y vertices: 5,

x

−8

−8

9. center: 1,

4

冉冊冉冊冉冊冉冊

1 2 1 1 vertices: 2, , 6, 2 2 1 foci: 2 兹7, 2

10. center: 2,

冉冉

12. vertex: (3, 2)

冉冊

focus: 3,

directrix: y 苷 1

[6.2]

y

冊冊

7 , 1 2 7 focus: , 3 2

11. vertex:

17 4

directrix: y 苷

[6.1]

y

1 4

[6.1]

y

8

8

−4

2 4

x

−8

4 −8

x

−2

x −8

−4 −8

4

8 x

A34

Answers to Selected Exercises

共x 2兲2 共 y 3兲2 共x 1兲2 共 y 1兲2 共x 2兲2 共 y 2兲2 苷 1 [6.2] 14. 苷 1 [6.3] 15. 苷 1 [6.3] 16. 共 y 3兲2 苷 8共x 4兲 [6.1] 25 16 9 7 4 5 3共 y 2兲共x 2兲2 共 y 1兲2 x2 y2 17. x 2 苷 or 共 y 2兲2 苷 12x [6.1] 18. 苷 1 [6.2] 19. 苷 1 [6.3] 20. y 苷共x 1兲2 [6.1] 2 9 5 36 4兾9 9兹2 兹2 21. 共x 兲2 2共y 兲2 4 苷 0, ellipse [6.4] 22. 6共x 兲2 x y 12 苷 0, parabola [6.4] 23. 共x 兲2 4y 8 苷 0, parabola [6.4] 2 2 1 1 24. 共x 兲2 共 y 兲2 兹2x 1 苷 0, hyperbola [6.4] 25. [6.5] 26. [6.5] 27. 2 2 13.

4

28.

[6.5]

29.

[6.5]

33.

[6.5]

6

4

[6.5]

34.

[6.5]

6

6

2

30.

31.

[6.5]

35. r sin2 u 苷 16 cos u [6.5]

[6.5]

4

36. r 4 cos u 3 sin u 苷 0 [6.5]

37. 3r cos u 2r sin u 苷 6 [6.5]

8

38. r 2 sin 2 苷 8 [6.5]

39. y 2 苷 8x 16 [6.5]

42. y 苷共tan 1兲x [6.5]

43.

[6.6]

40. x 2 3x y 2 4y 苷 0 [6.5] 44.

6

[6.6]

y 4

41. x 4 y 4 2x 2y 2 x 2 y 2 苷 0 [6.5] 45.

[6.6]

4 x

−4

2

48. y 苷 2x 1, x 1 [6.7]

49.

4

8 x −8

x2 y2 苷 1 [6.7] 16 9

y 4

y

[6.6]

y 8

3

−4

3 5 x [6.7] 4 2

46.

−8 1

47. y 苷

32.

6

4

50.

x2 y2 苷 1 [6.7] 1 16

y

y 4

2 x

4

4 x

−4

x

2

4 x

−4

−4

51. y 苷 2x, x 0 [6.7]

52. 共x 1兲2 共 y 2兲2 苷 1 [6.7] y 4

y

2

53. y 苷 2x , x 0 [6.7]

54.

[6.7]

7

y

2 4 −4

[6.5]

x

4 x

−4 −4

2

x

0 −1

200

A35

Answers to Selected Exercises 55.

[6.4]

15

−15

56.

[6.5]

3.1

−4.7

15

57.

[6.7]

300

4.7

−15

0

−3.1

2000 0

Max. height (nearest foot) of 278 feet is attained when t ⬇ 4.17 seconds.

Chapter 6 Quantitative Reasoning Exercises, page 447 QR1. They appear to be the same.

QR2. They appear to be the same.

QR3. 5.2 units

Chapter 6 Test, page 449 1. vertex: 共0, 0兲 focus: 共0, 2兲 directrix: y 苷 2 [6.1]

2.

3. vertices: (3, 4), (3, 6) foci: (3, 3), (3, 5) [6.2]

[6.2]

y

3 −4

6. vertices: (6, 0), (6, 0)

x2 (y 3)2 苷 1 [6.2] 45 9

5.

−4

4 x

7.

[6.3] 8. 73.15º [6.4]

y

9. ellipse [6.4]

4 −8

[6.5]

2

4

15. y 2 8x 16 苷 0 [6.5]

x

4

10. P(2,300º) [6.5]

4

x

−6

11.

[6.3]

y

4

−2

foci: (10, 0), (10, 0) 4x [6.3] asymptotes: y 苷

3

4.

12.

6

16. x 2 8y 16 苷 0 [6.6]

[6.5]

2

4

13.

[6.5]

6

2

17. (x 3)2 苷

1 y [6.7] 2

18.

14.

3

x2 (y 2)2 苷 1 [6.7] 16 1

y

y

6

6

3 −6

−3

3

6 x

−6

−3

3

−3

−3

−6

−6

6 x

冉

5 5兹3 , 2 2

冊

[6.5]

A36

Answers to Selected Exercises

19.

[6.7]

5

−1

20. 443 feet [6.7]

24π

−1

Xscl 苷 2

Cumulative Review Exercises, page 450 1. 2 兹10 [1.1]

7. 19 centimeters [2.2] 14. 298 feet [4.1] 19.

8. amplitude:

15. 38° [4.2]

π 2

3π 4

3. 共 g ⴰ f 兲共x兲苷 3 sin x 2 [1.5]

2. origin [1.4]

[6.5]

π 4

1 , period: 6 [2.5] 2

9. 3 [2.6]

16. 24.6i 17.2j [4.3]

4.

4 [2.1] 3

10. See [3.1].

6. tan共t兲苷兹3 [2.4]

5. 11 mph [2.1] 63 11. [3.2] 65

17. No. v w 苷 1 苷 0 [4.3]

12.

冉冊

18. 4 cis

2兹6 [3.5] 5

2 [5.2] 3

13.

5 [3.6] 13

20. y 苷 x2 2x 2 [6.7]

5

5π 4

7π 4

Exercise Set 7.1, page 461 1 ; g共3兲苷 1000 100 d. f 共x兲

1. f 共0兲苷 1; f 共4兲苷 81

3. g共2兲苷

15. a. k共x兲

c. h共x兲

17.

b. g共x兲

y

y 20

19.

5

5. h共2兲苷

9 8 ; h共3兲苷 4 27

y

23.

21.

1

x

−2

2

x

2

1 16

9. 9.19

11. 9.03

13. 9.74

y

2

2

10

7. j共2兲苷 4; j共4兲苷

x

x

2

25. Shift the graph of f vertically upward 2 units. 27. Shift the graph of f horizontally to the right 2 units. 29. Reflect the graph of f across the y-axis. 31. Stretch the graph of f vertically away from the x-axis by a factor of 2. 33. Reflect the graph of f across the y-axis and then shift this graph vertically upward 2 units. 35. Shift the graph of f horizontally to the right 4 units and then reflect this graph across the x-axis. 37. Reflect the graph of f across the y-axis and then shift this graph vertically upward 3 units. 12 6 39. no horizontal asymptote 41. no horizontal asymptote

−3

−4.7

3

4.7 −1

−6

A37

Answers to Selected Exercises 43. horizontal asymptote: y 苷 0

45. horizontal asymptote: y 苷 10

1 −1

12

8

0

−14

12

0

47. a. 442 million connections b. 2008 49. a. 233 items per month; 59 items per month b. The demand will approach 25 items per month. 51. a. 6400 bacteria; 409,600 bacteria b. 11.6 hours 53. a. 69.2% b. 7.6 55. a. 363 beneficiaries; 88,572 beneficiaries b. 13 rounds 57. a. 141 F b. after 28.3 minutes 59. a. 261.63 vibrations per second b. No. The function f 共n兲 is not a linear function. Therefore, the graph of f 共n兲 does not increase at a constant rate. 63. 65. 共, 兲 67. 关0, 兲 f(x) = e x y

2 x

2

y=x

69. Let f (x) 苷 e x and g(x) 苷 2x 5. Then h共 x兲苷 e 2x5 苷 f 关2x 5兴苷 f 关 g(x)兴苷 ( f ⴰ g)(x).

Prepare for This Section (7.2), page 467 PS1. 4

PS2. 3

PS4. f 1共x兲苷

PS3. 5

3x 2x

PS5. 兵x兩x 2其

PS6. the set of all positive real numbers

Exercise Set 7.2, page 476 1. 10 1 苷 10

3. 82 苷 64

19. ln y 苷 x 43.

5. 70 苷 x

21. log 100 苷 2

7. e4 苷 x

9. e0 苷 1

23. 2 苷 ln(x 5)

y

11. 10 2 苷 3x 1 27. 5

25. 2

31. 2

29. 3

y

45.

2

13. log3 9 苷 2

47.

2

33. 4

15. log4

1 苷 2 16

35. 12

37. 8

16

x

17. logb y 苷 x 39.

2 5

41.

10 3

y 4

2 4

8

12

16

x

48

−2

49.

96

144

x

51. 共3, 兲

y

8

−2

−2

−4

53. 共, 11兲

55. 共, 2兲共2, 兲

57. 共4, 兲

59. 共1, 0兲共1, 兲

61.

2

−2

63.

冉冊 7 , 3

3

11 , 2

x

6

65.

冉冊

y

67.

2

y

69.

4

y 6

4 4 −2

8

12

16

20

x

2

2 48

96

144

x

−2

12 4

8

16

20 x

A38

Answers to Selected Exercises

71. y

b. f 共x兲

73. a. k共x兲

c. g共x兲

d. h共x兲

5

75.

Y1=-21n(X)

2

5

9.4

0

10 x

−5 2.5

77.

0.5

79.

Y1=abs(1n(X))

−12

9.4

12

−0.5

−0.5 5

83.

Y1=1og(X+10)

10

0 0

1.5

81.

Y1=1og(Xˆ(1/3))

85. a. 2.0%

Y1=31og(abs(2X+10))

−14.4

− 0.5

b. 45 months

87. a. 3298 units; 3418 units; 3490 units

b. 2750 units

4.4

−2

89. 2.05 square meters

91. a. Answers will vary.

b. 96 digits

c. 3385 digits

d. 6,320,430 digits

93. f and g are inverse functions.

95. range of f: 兵 y兩1 y 1其; range of g: all real numbers

Prepare for This Section (7.3), page 480 PS1. ⬇ 0.77815 for each expression PS4. ⬇ 3.21888 for each expression

PS2. ⬇ 0.98083 for each expression PS5. ⬇ 1.60944 for each expression

PS3. ⬇ 1.80618 for each expression PS6. ⬇ 0.90309 for each expression

Exercise Set 7.3, page 489 1 1 1 log2 x 3 log2 y 7. log7 x log7 z 2 log7 y 9. 2 ln z 2 2 2 1 1 1 1 1 log4 z 2 3 log4 z log x log z ln z 11. 13. 15. 17. log关x 2共x 5兲兴 19. ln共x y兲 3 2 4 3 6 2 2 4 2 xy x y (x 4) (2x 5)兹w (x 3)y 3 3 21. log关 x 3 兹 y 共x 1兲兴 23. log 25. log6 27. ln 29. ln 31. ln z x2 x(x 2 3) yz 2 x3 1. logb x logb y logb z

3. ln x 4 ln z

5.

冉冊

33. 1.5395

35. 0.8672

37. 0.6131

3

45.

39. 0.6447

41. 8.1749

册

冋

册

43. 0.8735 4

49.

Y1=(1og(X–3))/(1og(8))

Y1=(1og((X–3)2))/(1og(3))

8.4 −1

−3

8.4

−2

Y1=-1og(abs(X–2))/log(5)

−2.7

6.7

−2

−1

8.4

−2

53. False; log 10 log 10 苷 2 but log共10 10兲苷 log 20 苷 2.

2

51.

冋

4

47.

Y1=(1og(X))/(1og(4))

−1

冉冊

55. True

冋

册

Answers to Selected Exercises 57. False; log 100 log 10 苷 1 but log共100 10兲苷 log 90 苷 1. 59. False;

A39

log 100 2 苷苷 2 but log 100 log 10 苷 1. log 10 1

61. False; 共log 10兲2 苷 1 but 2 log 10 苷 2. 63. 2 65. 500 501 67. 1⬊870,551; 1⬊757,858; 1⬊659,754; 1⬊574,349; 1⬊500,000 69. 10.4; base 71. 3.16 10 10 mole per liter 73. a. 82.0 decibels b. 40.3 decibels c. 115.0 decibels d. 152.0 decibels 75. 10 times as great 77. 5 79. 10 6.5I0 or about 3,162,277.7I0 81. 100 to 1 83. 10 1.8 to 1 or about 63 to 1 85. 5.5 89. a. M ⬇ 6 b. M ⬇ 4 c. The results are close to the magnitudes produced by using the amplitude-time-difference formula.

Prepare for This Section (7.4), page 494 PS1. log3 729 苷 6

PS2. 54 苷 625

PS3. loga b 苷 x 2

PS4. x 苷

4a 7b 2c

PS5. x 苷

3 44

PS6. x 苷

100共A 1兲 A1

Exercise Set 7.4, page 501 1. 6 21. 7

3.

3 2

23. 4

冉冊

5.

6 5

7. 3

25. 2 2兹2

log 70 log 120 log 315 3 11. 13. log 5 log 3 2 199 10 27. 29. 1 31. 3 33. 10 35. 2 95 9.

15. ln 10

17.

37. no solution

ln 2 ln 3 ln 6 39. 5

41. log共 20 兹401 兲

1 3 log 45. ln共 15 4兹14 兲 47. ln共 1 兹65 兲 ln 8 49. 1.61 51. 0.96 53. 2.20 55. 1.93 2 2 59. a. 8500, 10,285 b. in 6 years 61. a. 60 F b. 27 minutes 63. 3.7 years 65. 6.9 months 67. 6.67 seconds and 10.83 seconds 43.

120

0

57. 1.34

b. 48 hours c. P 苷 100 d. As the number of hours of training increases, the test scores approach 100%.

Percent score

69. a.

3 log 2 log 5 2 log 2 log 5

19.

120 0 Hours of training

1200

b. in 27 years, or the year 2026

c. B 苷 1000

Number of bison

71. a.

0

d. As the number of years increases, the bison population approaches, but never reaches or exceeds, 1000.

100 0 Years

250

b. 78 years

c. 1.9%

75. a. 1.72 seconds

b. v 苷 100

c. The object cannot fall faster than 100 feet per second.

Years

73. a.

0

1 0 Percent (written as a decimal) increase in consumption

77. 138 withdrawals

79. The second step. Because log 0.5 0, the inequality sign must be reversed.

81. x 苷

y y1

83. e0.336 ⬇ 1.4

A40

Answers to Selected Exercises

Prepare for This Section (7.5), page 506 PS1. 1220.39

PS3. 0.0495

PS2. 824.96

PS4. 1340

PS5. 0.025

PS6. 12.8

Exercise Set 7.5, page 516

7. a.

Micrograms of Na

1. a. 2200 bacteria

3. a. N共t兲 ⬇ 22,600e0.01368t

b. 17,600 bacteria

A

b. 3.18 micrograms

5. N(t) ⬇ 362,300 e0.011727t; 402,600

b. 27,700

c. ⬇15.07 hours

d. ⬇30.14 hours

9. ⬇6601 years ago

4 3 2 1 t

30 60 90 Time (in hours)

11. ⬇2378 years old 13. a. $9724.05 b. $11,256.80 15. a. $48,885.72 b. $49,282.20 c. $49,283.30 17. $24,730.82 19. 8.8 years ln 3 21. t 苷 23. 14 years 25. a. 1900 b. 0.16 c. 200 27. a. 157,500 b. 0.04 c. 45,000 29. a. 2400 b. 0.12 c. 300 r 5500 100 1600 31. P共t兲 ⬇ 33. P共t兲 ⬇ 35. a. $158,000, $163,000 b. $625,000 37. a. P共t兲 ⬇ 1 12.75e0.37263t 1 4.55556e0.22302t 1 4.12821e0.06198t 39. a. P共t兲 ⬇

b. 497 wolves 45. 3.1 years

8500 1 4.66667e0.14761t

41. a. 0.056

b. 0.98 second

1

49. a.

b. 2016

47. a. v 30 25 20 15 10 5 2

v

3

4

b. 42 F c. v 苷 32

c. after 54 minutes

43. a. 211 hours

d. As time increases, the velocity approaches, but never reaches or exceeds, 32 feet per second.

t

b. 2.5 seconds

c. ⬇24.56 feet per second

d. The average speed of the object was approximately 24.56 feet per second during the period from t 苷 1 to t 苷 2 seconds.

80 60 40 20 1

51. 45 hours

2

3

4

t

53. a. 0.71 gram

b. 0.96 gram

c. 0.52 gram

55. 2.91%

Prepare for This Section (7.6), page 521 PS1. decreasing

PS2. decreasing

PS3. 36

PS4. 840

PS5. 15.8

PS6. P 苷 55

Exercise Set 7.6, page 529 1. increasing exponential function

3. decreasing exponential function; decreasing logarithmic function

5. decreasing logarithmic function

32

3.2 4

−2

0 −1

4.7

0

7

7 −2.8

−1

b. 1386 hours

A41

Answers to Selected Exercises 7. y ⬇ 0.99628共1.20052兲x; r ⬇ 0.85705

9. y ⬇ 1.81505共0.51979兲x; r ⬇ 0.99978 11. y ⬇ 4.89060 1.35073 ln x; r ⬇ 0.99921 235.58598 2098.68307 13. y ⬇ 14.05858 1.76393 ln x; r ⬇ 0.99983 15. y ⬇ 17. y ⬇ 1 1.90188e0.05101x 1 1.19794e0.06004x 19. a. LinReg: P ⬇ 0.22129t 3.99190, r ⬇ 0.99288; ExpReg: P ⬇ 4.05326(1.04460)t, r 苷 0.99412 b. the exponential model c. $8.15 21. a. T ⬇ 0.06273共1.07078兲F b. 5.3 hours 23. a. T ⬇ 0.07881共1.07259兲F b. 7.5 hours; 2.2 hours 25. An increasing logarithmic model provides a better fit because of the concave-downward nature of the graph. 27. a. p ⬇ 7.862共1.026兲y b. 36 centimeters 29. a. LinReg: pH ⬇ 0.01353q 7.02852, r ⬇ 0.956627; LnReg: pH ⬇ 6.10251 0.43369 ln q, r ⬇ 0.999998. The logarithmic model provides a better fit. b. 126.0 31. a. p ⬇ 3200共0.91894兲t; 2012 b. No. The model fits the data perfectly because there are only two data points. 33. a. ExpReg: S ⬇ 7062.46390(0.956776)t, r ⬇ 0.89618; LnReg: S ⬇ 6995.50673 841.74326 ln t, r ⬇ 0.96240 b. the logarithmic model 11.26828 c. 4977 35. a. P(t) ⬇ b. 11 billion people 37. A and B have different exponential regression functions. 1 2.74965e0.02924t 39. a. ExpReg: y ⬇ 1.81120共1.61740兲x, r ⬇ 0.96793; PwrReg: y ⬇ 2.09385共x兲1.40246, r ⬇ 0.99999 b. The power regression function provides the better fit.

Chapter 7 Assessing Concepts, page 538 1. False; f 共x兲苷 x 2 does not have an inverse function. 2. True 5. False; j共x兲 is not an increasing function for 0 b 1. 6. c

3. True 4. False; h共x兲 is not an increasing function for 0 b 1. 7. b 8. f 9. a 10. e 11. d 12. g

Chapter 7 Review Exercises, page 539 1. 2 [7.2] 2. 4 [7.2] 3. 3 [7.2] y 12. 8 [7.4] 13. [7.1]

4. p [7.2]

5. 2 [7.4] 14. [7.1]

7. 3 [7.4] 15. [7.1]

6. 8 [7.4] y

8. 4 [7.4]

9. 1000 [7.4] 16. [7.1]

y

10. 10 10 [7.4]

11. 7 [7.4]

y

1 2

5

12 y

17. [7.1]

1

1 x

x y

18. [7.1]

y

19. [7.2]

1

x

4 2

1

1 y 2

y 5

x

2

x

x

4

22. [7.2] 2

x

3

x

y

20. [7.2]

1

4

21. [7.2]

5

−5

12

23. [7.1]

5 x − 4.7

−5

25. 43 苷 64 [7.2]

7

24. [7.1]

−4.7

26.

冉冊 1 2

4.7 −2 Xscl = 1 Yscl = 1

3

苷 8 [7.2]

27. 共兹2 兲4 苷 4 [7.2]

28. e0 苷 1 [7.2]

4.7

−7 Xscl = 1 Yscl = 1

29. log5 125 苷 3 [7.2]

30. log2 1024 苷 10 [7.2]

31. log10 1 苷 0 [7.2]

32. log8 2兹2 苷

1 [7.2] 2

1 1 1 logb x 2 logb y logb z [7.3] 35. ln x 3 ln y [7.3] 36. ln x ln y 4 ln z [7.3] 2 2 2 x5 xz 兹2xy 38. log [7.3] 39. ln [7.3] 40. ln [7.3] 41. 2.86754 [7.3] 42. 3.35776 [7.3] 共x 5兲2 z3 y 34.

33. 2 logb x 3 logb y logb z [7.3] 37. log共 x 2兹 x 1 兲 [7.3] 3

43. 0.117233 [7.3]

44. 0.578989 [7.3]

x

A42

45. 53. 61. 68. 72. 76.

Answers to Selected Exercises

1 ln 3 ln共 8 3兹7 兲 e [7.4] 49. 4 [7.4] 50. 15 [7.4] 51. [7.4] 52. [7.4] 6 2 ln 4 ln 5 15 兹265 [7.4] 57. 81 [7.4] 58. 兹5 [7.4] 59. 4 [7.4] 60. 5 [7.4] 10 1000 [7.4] 54. e共e 兲 [7.4] 55. 1,000,005 [7.4] 56. 2 7.7 [7.3] 62. 5.0 [7.3] 63. 3162 to 1 [7.3] 64. 2.8 [7.3] 65. 4.2 [7.3] 66. ⬇3.98 10 6 [7.3] 67. a. $20,323.79 b. $20,339.99 [7.5] a. $25,646.69 b. $25,647.32 [7.5] 69. $4,438.10 [7.5] 70. a. 69.9% b. 6 days c. 19 days [7.5] 71. N共t兲 ⬇ e0.8047t [7.5] N共t兲 ⬇ 2e0.5682t [7.5] 73. N共t兲 ⬇ 3.783e0.0558t [7.5] 74. N共t兲 ⬇ e0.6931t [7.5] 75. a. P共t兲 ⬇ 25,200e0.06155789t b. 38,800 [7.5] 340 years [7.5] 77. Answers will vary. [7.6] 78. a. ExpReg: R ⬇ 179.94943共0.968094兲t, r ⬇ 0.99277;

ln 30 [7.4] ln 4

46.

log 41 1 [7.4] log 5

47. 4 [7.4]

48.

2

LnReg: R ⬇ 171.19665 35.71341 ln t, r ⬇ 0.98574 b. The exponential equation provides a better fit for the data. 1400 1 births [7.6] 79. a. P共t兲 ⬇ b. 1070 coyotes [7.5] 80. a. 21 b. P共t兲 l 128 [7.5] 17 0.22458t 3 1 e 3

c. 5.4 per 1000 live

Chapter 7 Quantitative Reasoning Exercises, page 541 QR1. ExpReg: S ⬇ 5.94860共1.72192兲t; logistic: S ⬇ QR3. S ⬇ 58.73554 16.75801 lnt [7.6]

244.56468 [7.6] 1 39.38651e0.57829t

QR4. 88.8 million [7.6]

QR2. exponential: 1363.1 million; logistic: 218.1 million [7.6]

QR5. y ⬇

85.24460 , 84.2 million [7.6] 1 14.0040e0.70591t

QR6. Answers will vary. [7.6]

Chapter 7 Test, page 542 1. a. bc 苷 5x 3 [7.2] 5. [7.1]

b. log3 y 苷

y

x [7.2] 2

2. 2 logb z 3 logb y y

6. [7.2]

1 logb x [7.3] 2

2x 3 [7.3] 4. 1.7925 [7.3] 共x 2兲3 5 ln 4 8. [7.4] 9. 1 [7.4] 10. 3 [7.4] ln 28

3. log

7. 1.9206 [7.4]

1

5

5

x

5

−1

11. a. $29,502.36

b. $29,539.62 [7.5]

12. 17.36 years [7.5]

15. 690 years [7.5]

16. a. y ⬇ 1.67199共2.47188兲x

x

13. a. 7.6

b. 63 to 1 [7.3]

14. a. P共t兲 ⬇ 34,600e0.04667108t

b. 55,000 [7.5]

72.03783 17. a. LnReg: d ⬇ 67.35501 2.54015 ln t; logistic: d ⬇ 1 0.15279e0.67752 t 1100 18. a. P共t兲 ⬇ b. about 457 raccoons [7.5] 1 5.875e0.20429t

b. 1945 [7.6]

b. logarithmic: 73.67 meters; logistic: 72.03 meters [7.6]

Cumulative Review Exercises, page 543 1 3 4 3 x 4 [1.6] 3. sin 苷 , cos 苷 , tan 苷 [2.2] 4. 17 centimeters [2.2] 2 5 5 4 12 7 11 5. amplitude: 4, period: , phase shift: [2.5] 6. amplitude: 兹2, period: 2 [3.4] 7. odd [2.4] 9. [3.5] 10. , , [3.6] 4 5 2 6 6 11. magnitude: 5, direction: 126.9° [4.3] 12. 176.8° [4.3] 13. ground speed: ⬇420 mph; course: ⬇55 [4.1/4.2] 14. 26° [4.1] 15. 16 [5.3] 兹2 兹2 兹2 兹2 π π 16. [5.3] 17. 共兹2, 45° 兲 [6.5] 18. [6.5] 19. 1.43 [7.4] 20. 1.8 milligrams [7.5] i , i 2 3 π 2 2 2 2 1. cos共x2 1兲 [1.5]

2. f 1共x兲苷

4

4

π 6

Index Abscissa, 15 Absolute value of a complex number, 343–344 of a real number, 3 Absolute value function, 22, 34 Absolute value inequalities, 10–11 Acidity, 487–489 Acute angles, 119 Addition of complex numbers, 336 of functions, 69–70, 197 of ordinates, 197 of real numbers to inequalities, 7 of vectors, 313, 315, 316 Additive inverse, of a vector, 315 Adjacent side, 134, 135 Agnesi, Maria, 19 Air resistance, 206, 499–500, 515 Airspeed, 318 Alkalinity, 487–489 Amplitude of cosine functions, 174, 175 of simple harmonic motion, 204 of sine functions, 170–171, 172–173 Amplitude-Time-Difference Formula, 487 Analytic geometry, 15, 17 See also Conic sections Angle of depression, 139, 140 Angle of elevation, 139–140, 141 Angle of rotation, 404–405, 406 Angles classification of, 119–122 definition of, 118 degree measure of, 118–123, 124–126 naming of, 118 negative, 118 positive, 118 radian measure of, 123–126 reference angle of, 150–153 standard position of, 120 between vectors, 322 Angular speed, 127–129 Aphelion, 383 Appollonius, 363 Arc, 123, 124, 133 Archimedes, 2, 306 Arc length, 126–127 arcsin x, 256 Area of polygon, 146 of triangle, 305–307 Argand diagram, 342 Argument, of complex number, 344, 354 Asymptotes of hyperbolas, 392, 394 vertical, 38–39 Average velocity, 71–72

Axis, coordinate. See Coordinate axis Axis of symmetry, 54 –55, 57 of conic in polar coordinates, 427 of ellipse, 375 of parabola, 363, 364, 367 See also Symmetries of graphs Base of exponential function, 452– 453 of logarithmic function, 467, 482– 484 Bearing, 297 Beats, 202, 254 Benford’s Law, 479 Bernoulli, Johann, 437 Brachistochrone problem, 437– 438 Butterfly curve, 425 Calculators. See Graphing calculators Carbon dating, 508–509 Cardioid, 416, 417 Carrying capacity, 513 Cartesian (rectangular) coordinates, 15–16 rotation of, 404 transformation to/from polar coordinates, 419–423 Catenary, 466 Central angle, 123, 124 Change-of-base formula, 482–484 Circles arc length of, 126–127 arc of, 123, 124 as conic sections, 363, 375 definition of, 24 equations of, 24 –26, 407 great circle, 330–331 identifying with discriminant, 407–408 involute of, 442 parametric form of, 435 polar equations of, 413, 415, 432 sector of, 133 unit circle, 155–157, 159–160 Circular functions, 157 cis notation, for complex number, 344 Closed intervals, 2, 3 Coefficient of determination, 100, 101, 102, 524, 525, 527 Cofunction identities, 227–228 Common logarithms, 473, 474 Complementary angles, 120 Completing the square, 26 Complex conjugates, 338–339 Complex numbers, 335–340 absolute value of, 343–344 addition of, 336 argument of, 344, 354 division of, 338–339, 347, 348 graphs of, 342, 343

modulus of, 344 multiplication of, 337–338, 346, 348 powers of, 350–351 roots of, 352–354, 443 standard form of, 343 trigonometric form of, 343–348 Complex plane, 342, 343 Components of a vector, 314, 315, 317–318, 319 Composition of functions, 72–76 of function with its inverse, 84–85 logarithmic and exponential, 468 trigonometric, 259–263 Compound interest, 509–512 Compressing graphs, 61–63 Computer algebra systems, 389 Concavity, 522–523, 526 Conic sections, 363, 374 –375, 390 degenerate, 375, 390, 403, 407 focus-directrix definitions of, 426–428 general equation of, 403, 407 graphing with calculator, 408–409 identification theorem for, 407–408 polar equations of, 426–430 rotation of, 403– 407 See also Circles; Ellipses; Hyperbolas; Parabolas Conjugate of complex number, 338–339 of trigonometric expression, 219–220 Conjugate axis, of hyperbola, 391 Constant functions, 40 Continuous functions, 41 Conversion factors, 122 Coordinate axis polar, 412 real number line, 2 rotation of, 404 symmetry with respect to, 54–55, 57 Coordinate plane, 15, 16 Coordinates of point in a plane, 15–16, 412–413 of point on real number line, 2 See also Cartesian (rectangular) coordinates; Polar coordinates Correlation coefficient, 99–100, 524, 525, 527 Cosecant of acute angles, 134–135 of any angle, 147–148 domain and range of, 160 graph of, 185–187, 196 inverse function of, 257, 258–259 as odd function, 160 period of, 161–162 of quadrantal angles, 148–149 of a real number, 157–158 sign of, 149 of special angles, 136–138

I1

I2

Index

Cosine of acute angles, 134–135 of any angle, 147–148 approximating with calculator, 139 domain and range of, 159, 160 dot product and, 321–322 of double angle, 237 as even function, 160 graph of, 173–177 graph translations of, 192–193, 195–196 of half angle, 240–241 inverse function of, 256–257, 259–263 Law of Cosines, 302–305 period of, 161, 162 of quadrantal angles, 148–149 of a real number, 157–162 reference angle and, 152 sign of, 149 of special angles, 136–138 of sum or difference of angles, 226–227, 231 Cotangent of acute angles, 134–135 of any angle, 147–148 domain and range of, 160 graph of, 184–185 graph translations of, 194 inverse function of, 257, 258, 259 as odd function, 160 period of, 162 of quadrantal angles, 148–149 of a real number, 157–158 sign of, 149 of special angles, 136–138 Coterminal angles, 121–122 Critical value method, 8–10 Cryptography, 114–115 Cube roots, 352 Cubic equations, 5 Curve orientation of, 437 parametric equations for, 433–434 Cycle, of sine graph, 170 Cycloid, 437–438 Damped harmonic motion, 206–207 Damping factor, 198 Day of the week, 53 Decibels (dB), 491 Decimal degree method, 122–123 Decimals, 2 Decreasing functions, 40, 82 Degree measure, 118–123 conversion to/from radians, 124 –126 de Moivre, Abraham, 351 De Moivre’s Theorem, 350–354 Dependent variable, 32 Descartes, René, 15, 335 Difference. See Subtraction Difference quotient, 70–72 Direction angle, 315, 317 Directrix of conic sections, 426–428 of parabola, 363, 364, 367

Dirichlet, Peter Gustav, 68 Dirichlet function, 68 Discontinuities, 41, 42 Discriminant, of second-degree equation in two variables, 407–408 Displacement in simple harmonic motion, 204 as vector quantity, 313 work and, 324 Distance in Cartesian coordinates, 17 between points on a line, 3, 17 in polar coordinates, 425 speed and, 128 See also Displacement Distance formula, 17 Division of complex numbers, 338–339, 347, 348 of functions, 69–70 of inequalities by real numbers, 7 DMS (Degree, Minute, Second) method, 122–123 Domain of function, 32, 36 of inverse function, 82 of sum or product of functions, 69–70 Dot product, 320–324 Double-angle identities, 236–238 e (base of natural exponential function), 458 Earthquakes, 484–487, 492–493 Eccentricity of conic sections, 426–428 of ellipse, 381–382 of hyperbola, 396 Ellipses, 363, 374–384 applications of, 382–384 with center at (0, 0), 375–378 with center at (h, k), 378–381 definition of, 374–375 degenerate, 375 eccentricity of, 381–382 general equation of, 403, 405–406, 407 identifying with discriminant, 407–408 latus rectum of, 389 parametric form of, 436 polar equations of, 426– 428, 429 reflective property of, 384 Endpoint, of ray, 118 Epicycloid, 442 Epitrochoid, 448–449 Equality of ordered pairs, 17 of real numbers, 2 Equality of Exponents Theorem, 495 Equations cubic, 5 definition of, 4 exponential, 494–497 functions represented by, 32 linear in one variable, 4 literal, 4 –5 logarithmic, 497–500, 512 parametric, 432–440, 447– 449

polar, 413– 419, 421– 423, 426– 430 quadratic, 5–6 quartic, 5 second-degree in two variables, 403, 407 trigonometric, 262, 268–275, 288–289, 328 in two variables, 19–26 See also Solutions Equilibrium point, 204 Equivalent inequalities, 7 Equivalent vectors, 313 Euler, Leonhard, 2, 32, 458 Evaluating a function, 33, 70 Even and odd functions, 56–57 trigonometric, 160–161 Exponential decay, 506, 508–509, 535–536 Exponential equations, 494–497 Exponential functions, 452–461 definition of, 452–453 evaluating, 453 graphs of, 453–457 logarithmic functions and, 467–469 models based on, 451, 459–461, 521–527 natural, 458– 459 properties of, 455 Exponential growth, 506–507, 511, 512 Exponential regression, 524, 525, 527 Exponents equality of, 495 See also Powers Factoring quadratic equations, 6 Falling objects, 499–500, 515 Family of curves, 58–59, 210 Fermat, Pierre de, 15 Floor function, 41– 44 Focus of conic sections, 426–428 of ellipse, 375, 376, 377, 379 of hyperbola, 390, 391, 394, 396 of parabola, 363, 364, 367, 369 Force vectors, 318, 319–320, 323–324 Formulas, 45 Fractals, 334, 341 Fractions as conversion factors, 122 as rational numbers, 2 Frequency, of harmonic motion, 204, 205, 207 Functions, 31–32 algebraic operations on, 69–70, 197 composition of, 72–76, 84–85 continuous, 41 definition of, 32 difference quotient of, 70–72 domain of, 32, 36 evaluating, 33, 70 even and odd, 56–57 families of, 58–59, 210 graphs of, 37–41 identifying, 35–36 inverse, 81–90 notation for, 33 one-to-one, 40–41, 82, 84, 86 piecewise-defined, 33–35, 108–109

Index range of, 32 vertical line test for, 39– 40 Golden rectangles, 14 Graphing calculators adjusting settings, 44 angle measure, 123, 125, 126 complex numbers, 338 connected mode, 42 dot mode, 42, 109 EPIT program, 448 estimating x-intercepts, 23–24 evaluating exponential expressions, 458 evaluating inverse trigonometric functions, 283–284 evaluating logarithms, 474 evaluating trigonometric functions, 138–139 finding nth roots of complex number, 443 graphing absolute value equation, 22 graphing Colosseum model, 389 graphing conic sections, 408–409 graphing cycloid, 438 graphing damped harmonic motion, 207 graphing ellipses, 380 graphing equations in two variables, 22 graphing exponential functions, 459, 460 graphing families of curves, 58–59, 210 graphing functions, 37, 42, 44 graphing hyperbolas, 395 graphing inverse functions, 85–86, 90 graphing inverse trigonometric functions, 263, 264 graphing logarithmic functions, 474, 484 graphing Mandelbrot set, 359–360 graphing parabolas, 366 graphing parametric equations, 438, 439 graphing piecewise functions, 108–109 graphing polar equations, 417, 419 graphing second-degree equations in two variables, 408–409 greatest integer function, 41, 42 holes in graphs, 68 isolated points in graphs, 68 LIST feature, 58–59 logistic models, 528–529 Mandelbrot iteration procedure, 356–357 median–median line, 108 modeling guidelines, 526–527 POWERMOD program, 115 REFANG program, 151 regression analysis, 97–98, 101–102, 275–277, 524, 525, 527 ROTATE program, 408 solving exponential equations, 495–497 solving trigonometric equations, 273–275, 328 SQUARE viewing window, 86 TABLE feature, 20–21 viewing window, 85–86 WANKEL program, 447–448 ZERO feature, 23–24 Graphs of circles, 413, 415

of complex numbers, 342, 343 compressing, 61–63 of conic sections, 403, 405–408, 426–429 of cosecant functions, 185–187, 196 of cosine functions, 173–177, 192–193, 195–196 of cotangent functions, 184–185, 194 of difference of functions, 197 of ellipses, 375–376, 427, 429 of equations in two variables, 19–22 of even and odd functions, 57 of exponential functions, 453–457 of floor function, 41–42 of functions, 37–41 of hyperbolas, 391–392, 427, 428–429 intercepts of, 22–24 of inverse functions, 83–84, 85–86, 90 of inverse trigonometric functions, 263 of logarithmic functions, 470–472 of parabolas, 22, 364, 366, 367, 427, 430 of parametric curves, 433–434 of polar equations, 413–419, 427, 428–429 reflections of, 60 scatter diagrams, 16 of secant functions, 187–189 semilog, 535–536 of sets of real numbers, 2–3 of sine functions, 169–173 stretching, 61–63 of sum of functions, 197 of tangent functions, 180–183 translations of, 58–59, 60 See also Symmetries of graphs Great circle distance, 330–331 Greater than, 2 Greatest integer function, 41–44 Ground speed, 318 Growth models exponential, 506–507, 511, 512 logistic, 513–514, 528–529 Half-angle identities, 240–242 Half-life, 508 Half-line, 118 Half-open intervals, 2, 3 Harmonic motion damped, 206–207 simple, 204–206 Head, of a vector, 313 Heading, 297, 331 Heron’s formula, 306–307 Herschel, Caroline, 397 Holes in graphs, 68 Horizontal component, 317–318 Horizontal compressing and stretching, 62–63 Horizontal lines, 413 Horizontal line test, 40–41 Horizontal translations, 58, 59 Hyperbolas, 363, 390–398 applications of, 396–398 asymptotes of, 392, 394 with center at (0, 0), 391–393 with center at (h, k), 393–395

I3

definition of, 390 degenerate, 390 eccentricity of, 396 general equation of, 403, 407 identifying with discriminant, 407–408 polar equations of, 426–429 reflective property of, 398 Hypocycloid, 442 i (imaginary unit), 335 powers of, 339–340 i (unit vector), 316 Identities, trigonometric. See Trigonometric identities Imaginary axis, 343 Imaginary numbers, 335, 336 Imaginary part, 335 Imaginary unit, 335, 339–340 Increasing functions, 40, 82 Independent variable, 32 Inequalities, 6–8 with absolute values, 10–11 equivalent, 7 with polynomials, 8–10 properties of, 7 solution sets of, 6–7, 8, 10 solving by critical value method, 8–10 Inequality symbols, 2 Infinity, approach of function to, 38 Initial point, of vector, 313 Initial side, of angle, 118 Inner product, 320–324 Integers, 2 Intercepts, 22–24 Interest compound, 509–512 simple, 509 Internet, 1 Interval notation, 2–3 Inverse functions, 81–83 composition of a function with, 84–85 finding, 86–89 graphs of, 83–84, 85–86, 90 logarithmic and exponential, 468 with restricted domain, 88 trigonometric, 255–265, 283–284 Inverse relations, 82, 84 Inverse trigonometric functions, 255–265 approximating with polynomials, 283–284 composition with, 259–263 equations with, 262 graphing with calculator, 263, 264 identities with, 259, 262–263 Involute of a circle, 442 Irrational numbers, 2, 458 Isolated points in graphs, 68 Iterates, 341, 356 Iteration, 341 Mandelbrot procedure, 356–357 j (unit vector), 316 Kepler’s Laws, 389

I4

Index

Latitude, 330–331 Latus rectum of ellipse, 389 of parabola, 373 Law of Cosines, 302–305 Law of Sines, 293–298, 304 Least-squares regression line. See Linear regression Less than, 2 Libby, Willard Frank, 509 Limaçon, 416–417 Line(s) graphing an equation of, 19 parametric forms of, 435 polar equations of, 413, 432 symmetry with respect to, 54–55, 57, 414, 415, 416 Linear correlation coefficient, 99–100 Linear equations in one variable, 4 Linear regression, 95–100, 102 Linear speed, 127–129 Line of best fit. See Linear regression Line segment, midpoint of, 18 Lissajous figures, 442 Literal equations, 4–5 Logarithmic decrement, 209 Logarithmic equations, 497–500, 512 Logarithmic functions, 467– 475 applications of, 474–475, 484–489, 492–493 common, 473–474 definition of, 467 domains of, 472–473 exponential functions and, 467–468 graphs of, 470–472, 474, 484 modeling data with, 521–522, 523, 524, 526–527 natural, 473–474 properties of, 472 Logarithmic regression, 524, 526–527 Logarithmic scales, 484–489, 492– 493 Logarithms change-of-base formula for, 482–484 changing to exponential form, 468– 469 common, 473, 474 definition of, 467 natural, 473, 474 properties of, 469–470, 480–482 Logistic models, 513–514, 520–521, 528–529 Longitude, 330–331 Mach numbers, 245 Magnitude of a vector, 313, 315, 321 Major axis, 375, 376, 377, 379, 381 Mandelbrot, Benoit B., 334 Mandelbrot iteration procedure, 356–357 Mandelbrot set, 357, 359–360 Maple, 389 Maximum value, 186 Measure of an angle, 118 in degrees, 118–123, 124–126 in radians, 123–126 Median–median line, 108

Midpoint formula, 18 Minimum value, 186 Minor axis, 375, 376 Minute (angle measure), 122–123 Modeling data, 524 –529 linear, 95–100, 102 quadratic, 100–102 sinusoidal, 275–277 Modular functions, 1, 114 –115 Modulus, of complex number, 344 Mollweide’s formulas, 311 Motion of falling objects, 499–500, 515 parametric equations for, 436–437 projectile, 273–275, 439 vibratory, 204 –207 See also Force vectors; Speed Multiplication of complex numbers, 337–338, 346, 348 of functions, 69–70 of inequalities by real numbers, 7 of vectors (dot product), 320–324 of vectors (scalar), 313, 315, 316 Napier, John, 467 Natural exponential function, 458– 461 Natural logarithms, 473, 474 Natural numbers, 2 Negative angles, 118 Never-negative function, 80–81 Nomograms, 492– 493 Oblique triangle, 293, 304 Obtuse angle, 119 Odd and even functions, 56–57 trigonometric, 160–161 One-to-one function, 40– 41 as inverse function, 82, 84, 86 Open intervals, 2, 3 Opposite side, 134, 135 Ordered pairs as coordinates, 15, 16, 412–413 equality of, 17 of a function, 32, 37 graph of, 37 of inverse function, 82 of a relation, 31 as solutions of equation, 19 Ordinate, 15 Orientation of curve, 437 Origin, 15 symmetry with respect to, 55–56, 57 Orthogonal vectors, 323 Oscillation. See Harmonic motion Parabolas, 363–370 applications of, 368–370 definition of, 363 general equation of, 403, 407 graphs of, 22, 364, 366, 367, 427, 430 identifying with discriminant, 407–408 latus rectum of, 373 parametric form of, 434

polar equations of, 426– 428, 429–430 reflective property of, 369 with vertex at (0, 0), 363–366 with vertex at (h, k), 366–368 vertex of, 363, 364, 367, 369 Paraboloid, 369 Parallelogram method, of vector addition, 313 Parallel vectors, 323 Parameter, of family of functions, 58 Parametric equations, 432– 440 for rotary engine, 447– 449 Perihelion, 383 Period of cos bx, 174, 175 of cot bx, 184 of csc bx, 186 definition of, 161 of harmonic motion, 204 of sec bx, 188 of sin bx, 172 of sum of functions, 214 of tan bx, 182 of trigonometric functions, 161–162 Periodic functions, 161–162 Perpendicular vectors, 323 Phase shift, 192–193, 194, 249–250 pH of a solution, 487–489 Pi (), 2 Piecewise-defined functions, 33–35 graphing with calculator, 108–109 Plotting points, 16 in graphing a function, 37–38 in graphing an equation, 20, 21–22 Point(s) of coordinate plane, 15 of real number line, 2 symmetry with respect to, 55–56, 57, 414 See also Plotting points Poiseuille, Jean Louis, 328 Poiseuille’s Law, 328 Polar axis, 412 Polar coordinates, 412– 413 distance in, 425 symmetries in, 414, 415, 416, 427 transformation to/from rectangular coordinates, 419– 421 Polar equations, 413 of conic sections, 426–430 graphs of, 413–419 writing as rectangular equations, 421–423 Polar form, of complex number, 344 Pole, 412, 414 Polygons, areas and perimeters of, 146 Polynomials approximating inverse trigonometric functions with, 283–284 inequalities with, 8–10 sign property of, 8 zeros of, 8 Population growth, 507, 513–514 Positive angles, 118 Power functions, 534

Index Power-reducing identities, 239 Powers of a complex number, 350–351 Equality of Exponents Theorem, 495 of i, 339–340 Principal, 509 Product. See Multiplication Product-to-sum identities, 246–247 Projectile motion, 273–275, 439 Projection, scalar, 322–323 Protractor, 119, 124 Pseudoperiod, 207 Public key cryptography, 114 –115 Pythagorean identities, 163, 217, 218–219 Pythagorean Theorem, 17 Quadrantal angles, 121 trigonometric functions of, 148–149 Quadrants, 15 Quadratic equations, 5–6 Quadratic formula, 5–6 Quadratic regression, 100–102 Quartic equations, 5 Quotient. See Difference quotient; Division Radian measure, 123–126 Radioactive decay, 508–509 Radius of a circle, 24, 25, 26 Range of a function, 32 of an inverse function, 82 of trigonometric functions, 159–160 Ratio identities, 163, 217 Rational numbers, 2 Ray, 118 Real axis, 343 Real number line, 2–3 Real numbers, 2 Real part, of complex number, 335 Reciprocal functions, 138 Reciprocal identities, 163, 217 Rectangular (Cartesian) coordinates, 15–16 rotation of, 404 transformation to/from polar coordinates, 419–421 Rectangular form, of complex number, 343 Reduction formulas, trigonometric, 232 Reference angle, 150–153 Reflections of graphs, 60 of exponential functions, 457 Reflective property of ellipse, 384 of hyperbola, 398 of parabola, 369 Regression analysis, 524 –529 exponential, 524, 525, 527 linear, 95–100, 102 logarithmic, 524, 526–527 quadratic, 100–102 sine, 275–277 Relations, 31 inverse of, 82, 84

Relative maximum, 186 Relative minimum, 186 Resolving a vector, 319 Resultant vector, 313 Richter, Charles F., 484 Richter scale, 484– 487 Right angles, 119 Right triangles applications of, 139–141 Pythagorean Theorem, 17 trigonometric expressions evaluated with, 260–263 trigonometric functions defined with, 134–138 Roots of a complex number, 352–354, 443 of an equation, 4 See also Solutions; Zero of a polynomial Rose curves, 418 Rotary engine, 447–449 Rotation of axes, 404 Rotation Theorem for Conics, 404– 407 Rounding numbers, 42–43, 140 Scalar, 312 Scalar multiplication, of a vector, 313, 315, 316 Scalar product, of two vectors, 320–324 Scalar projection, 322–323 Scalar quantities, 312 Scatter diagrams, 16, 524, 526 See also Regression analysis Secant of acute angles, 134–135 of any angle, 147–148 domain and range of, 160 as even function, 160 graph of, 187–189 inverse function of, 257, 258, 259 period of, 161–162 of quadrantal angles, 148–149 of a real number, 157–158 sign of, 149 of special angles, 136–138 Second (angle measure), 122–123 Sector, 133 Seed, 341, 356 Seismograms, 486–487, 492–493 Semilog graphs, 535–536 Semimajor or semiminor axis, 375 Semiperimeter, 307 Set-builder notation, 2–3 Sets of real numbers, 2–3 Sign diagram, 9 Significant digits, 140 Simple harmonic motion, 204 –206 Simple interest, 509 Sine of acute angles, 134–135 of any angle, 147–148 domain and range of, 159, 160 of double angle, 236, 237–238 graph of, 169–173

I5

graph translations of, 192–193, 194 of half angle, 240 inverse function of, 255–256, 259–262, 263, 283–284 Law of Sines, 293–298, 304 modeling data with, 275–277 as odd function, 160 period of, 161, 162 of quadrantal angles, 148–149 of a real number, 157–158 reference angle and, 152 sign of, 149–150 of special angles, 136–138 of sum or difference of angles, 226, 229, 230, 232 Sinusoidal data, 275–277 Solutions of an equation in one variable, 4 of an equation in two variables, 19 extraneous, 270, 498–499 of a polar equation, 413 of a trigonometric equation, 268–269, 271, 273 Solution set of an inequality, 6–7, 8, 10 Sørenson, Søren, 488 Sound beats, 202, 254 decibels, 491 whispering galleries, 383–384 Sound tracks, 214 Special angles, 136–138 Speed airspeed vs. ground speed, 318 angular vs. linear, 127–129 Mach number for, 245 as magnitude of velocity, 313 See also Velocity Spring constant, 205 Square roots, of negative numbers, 335 Step function, 42 Straight angle, 119 Stretching graphs, 61–63 of exponential functions, 457 Subtending an angle, 123, 124 Subtraction of complex numbers, 336 of functions, 69–70, 197 of real numbers from inequalities, 7 of vectors, 313 Sum. See Addition Sum-to-product identities, 247–249 Supplementary angles, 120 Symmetries of graphs of function and its inverse, 84, 85 in polar coordinates, 414, 415, 416, 427 with respect to coordinate axis, 54–55, 57 with respect to line, 54 –55, 57, 414, 415, 416 with respect to point, 55–56, 57, 414 See also Axis of symmetry Tail, of vector, 313

I6

Index

Tangent of acute angles, 134–136 of any angle, 147–148 domain and range of, 159, 160 of double angle, 237 graphs of, 180–183 graph translations of, 194 of half angle, 240, 241 inverse function of, 257, 258, 259, 260, 264 as odd function, 160 period of, 162 of quadrantal angles, 148–149 of a real number, 157–158 reference angle and, 152 sign of, 149–150 of special angles, 136–138 of sum or difference of angles, 226, 229, 230 Tangent lines, concavity and, 522 Terminal point, of vector, 313 Terminal side, of angle, 118 Test value, 8–9 Time as parameter, 436–437 speed and, 128 Transformation equations, 366, 378, 393 Translations of graphs, 58–59, 60 of conic sections, 366, 378, 393, 403 of exponential functions, 456–457 of inverse trigonometric functions, 263 of logarithmic functions, 473 of trigonometric functions, 192–196 Transverse axis, of hyperbola, 391, 394, 396 Triangle method, of vector addition, 313 Triangles area of, 305–307 checking solutions with Mollweide’s formulas, 311 oblique, 293, 304 solving with Law of Cosines, 302–305 solving with Law of Sines, 293–298, 304 See also Right triangles Trigonometric equations, 268–275 applications of, 273–275, 288–289, 328 with inverse functions, 262 Trigonometric expressions, evaluating, 260–261 Trigonometric form, of complex numbers, 343–348 Trigonometric functions of acute angles, 134–136

of any angle, 147–148 approximating with calculator, 138–139 composition with inverses, 259–263 domain and range of, 159–160 even and odd, 160–161 graphs of, 169–189 inverse, 255–265, 283–284 periodic properties of, 161–162 properties of, 159–162 of quadrantal angles, 148–149 of real numbers, 157–162 reciprocal, 138 reference angles and, 153 signs of, 149–150 of special angles, 136–138 translations of, 192–196 Trigonometric identities cofunctions, 227–228 double-angle, 236–238 fundamental, 163–165, 217 half-angle, 240–242 with inverse functions, 259, 262–263 odd-even, 160, 217 power-reducing, 239 product-to-sum, 246–247 Pythagorean, 163, 217, 218–219 ratio, 163, 217 reciprocal, 163, 217 reduction formulas, 232 sum of sines and cosines, 249–251 sum or difference of two angles, 226–227, 229–231 sum-to-product, 247–249 verification of, 217–222 Unit circle, 155–157, 159–160 Unit fraction, 122 Unit vectors, 315–318 Variable dependent, 32 independent, 32 Vectors, 312–324 addition of, 313, 315, 316 additive inverse of, 315 angle between, 322 applications of, 318–320, 323–324 components of, 314, 315, 317–318, 319 definition of, 313 direction angle of, 315, 317

dot product of, 320–324 equivalent, 313 magnitude of, 313, 315, 321 orthogonal, 323 parallel, 323 scalar multiplication of, 313, 315, 316 scalar projection of, 322–323 subtraction of, 313 unit vectors, 315–318 zero vector, 315 Velocity average, 71–72 as vector, 313, 318 See also Speed Vertex of angle, 118 of ellipse, 375, 376, 377, 379 of hyperbola, 391, 394 of parabola, 363, 364, 367, 369 Vertical asymptote, 38–39 Vertical component, 317–318 Vertical compressing and stretching, 61 Vertical lines, 413 Vertical line test, 39–40 Vertical translations, 58–59, 60 Vibratory motion, 204 –207 Wankel, Felix, 447, 448 Wessel, Caspar, 342 Whispering galleries, 383–384 Work, force and, 323–324 World Wide Web, 1 Wrapping function, 155–157 x-axis, 15 reflection across, 60 symmetry with respect to, 54 –55 x-coordinate, 15 x-intercepts, 22–24 xy-plane, 15, 16 y-axis, 15 reflection across, 60 symmetry with respect to, 54 –55, 57 y-coordinate, 15 y-intercept, 22–23 Zeller’s Congruence, 53 Zero of a polynomial, 8 Zero product principle, 6 Zero vector, 315

Index of Applications • Agriculture

• Automotive

Area of a pasture, 310 Crop yield, 51 Dimensions of a fenced region, 13 Dimensions of a garden, 13

Braking distance, 106 Fuel efficiency, 106 Headlight design, 373 Movement of a bus, 282 Speed of a car, 131

• Architecture and Art Colosseum in Rome, 389 Eiffel Tower, 502 Eiffel Tower replica, 145 Golden Gate Bridge cables, 113 London bridge clearance, 388 Observation angle, 282 Petronas Towers, 145 Saint Louis Gateway Arch, 466 Washington Monument, 145

• Astronomy and Space Science Altitude of the sun, 281 Apparent magnitude of stars, 477 Cepheid variable stars, 179 Diameter of the sun, 131 Dimensions of telescopes, 372 Distance from Earth to Jupiter, 144 Focus of a radio telescope, 372 Focus of an optical telescope, 372 Height of the Eiffel Tower, 143 Height of the space shuttle, 141, 299 Kepler’s Laws, 389 Moons of Saturn, 283 Neptune, 389 Orbits of planets/comets, 144, 382, 383, 387, 388, 396, 397 Path of a satellite, 167 Percent of the moon illuminated, 275, 280 Period-luminosity relationship, 179 Radius of Earth, 132 Recession velocities, 107 Rotation time of the space shuttle, 133 Satellite coverage, 266 Velocity of the Hubble Space Telescope, 132

• Construction

Animal populations, 464, 503, 513, 517, 518, 520, 528, 533, 541, 543 Bacteria, 104, 516 Bertalanffy’s equation, 502 Biologic diversity, 493 Drag resistance on a fish, 224 Flying speeds, 105 Larvae population, 106 Logistic growth model, 513, 517 Mosquito population, 81 Oxygen consumption of a bird, 107 Predator-prey interactions, 203

Area of a housing tract, 310 Construction of a box, 45 Cooling tower design, 401, 402 Cost of glass for the Luxor Hotel, 307 Cross-sectional area of a rain gutter, 281 Dimensions of trusses, 311 Inclination angle of a ramp, 326, 330 Length of a brace, 300 Length of a guy wire, 51, 300 Length of a street, 309 Length of an airport runway, 299 Fountain design, 371 Placement of a light, 143 Semielliptical plywood form, 388 Vent pipe design, 386 Whispering gallery, 384, 387

• Business and Economics

• Consumer Applications

Cellular telephone subscriptions, 540 Cost and sales or profit, 49, 52 Cost of a commercial lot, 310 Demand for a product, 463 Depreciation, 518, 540 Digital camera sales, 541 Money market rates, 477 Monthly revenue, 14 Postage rates, 48, 92 Retail/wholesale price of a bracelet, 89 Revenue, 16, 460, 517 Sales, 27, 92, 200, 463, 517 Telecommuting, 531

Clothing sizes, 92 Cost of a car rental, 13 Cost to paint a house, 52 Cost of parking, 43 Depreciation of a boat, 49 Photochromatic eyeglass lenses, 463 Value of a car over time, 44 Value of a diamond, 524 Video rental plans, 13

• Archeology Age of a bone, 516 Age of the Rhind Papyrus, 516 Height of the Cheops pyramid, 144 Pterosaur wingspan data, 103

Speed of a disk, 128 Time to download a file, 113

• Biology and Life Sciences

• Chemistry Henderson-Hasselbach function, 532 Hydronium-ion concentration, 489, 491, 540 Absorption of oxygen, 532 pH of a solution, 487, 488, 491, 540 Viscosity of oil, 535

• Computers and Computer Science Animated maps, 491 Cryptology, 93, 94 Internet connections, 463 Moore’s Law, 452 Public key cryptology, 114

• Earth Science Amplitude-time distance formula, 487 Earthquakes, 484, 485, 486, 491, 492, 540, 542 Richter scale, 484, 492, 540, 542 Seismogram, 486, 492, 540

• Ecology and Demographics Carbon dioxide levels, 200 Generation of garbage, 530 Natural resources, 504 Oil spill, 72, 520 Population growth of Aurora, Colorado, 516 Hawaii, 532 Miami, Florida, 516 Population of a city, 501, 507, 516, 520, 540, 542 World population, 532, 534

• Entertainment Amusement park ride, 131, 209 Elliptical pool table, 387 Horse racing, 131 Linear speed on a carousel, 213 Millennium Wheel, 158 Movie ticket prices, 530 Number of cinema sites, 533 Photography, 132

• Finance Compensation, 92 Compound interest, 509, 510, 511, 512, 516, 517, 540, 542 Cost of a checking account, 13 Currency conversion, 133 Income tax, 48 Payment plans, 13 Per capita income, 27 Retirement planning, 504

• General Interest 3-D optical illusion, 374 Average rate of change, 79 Benford’s Law, 479 Birthday problem, 92 Cubic formula, 14 Dirichlet function, 68 Exponential reward, 466 Heron’s formula, 306 Mandelbrot iteration procedure, 356 Median-median line, 108 Mollweide’s formulas, 311 Never-negative function, 80 Nomogram, 492 Number of digits in bx, 478 Pay It Forward model, 464 Zeller’s Congruence, 53

• Geography Latitude, 133

• Geometry Epicycloid, 442 Epitrochoid, 448 Height of an inscribed cylinder, 50 Hypocycloid, 442 Involute of a circle, 442 Perimeter of a scaled rectangle, 76, 78, 79

• Medicine, Health, and Nutrition Anesthesiology, 477 Average remaining lifetime, 105 Blood pressure, 195 Calories, 106, 116

Concentration of a medication, 79, 93, 463, 516, 520 E. Coli infection, 463 Lithotripsy, 386 Mortality rate, 541 Optimal branching of arteries, 282, 328 Optometry, 532 Skin wound, 540

• Meteorology Average high temperature in Fairbanks, Alaska, 167 Distance from a hurricane to a city, 298 Hours of daylight, 201, 281, 290 Minimum-maximum temperature range, 13 Sunrise/sunset times, 279, 280, 288 Temperatures, 106, 202 Tidal cycle for Gray’s Harbor, 208 Tides, 202

• Miscellaneous Angle between the boundaries of a lot, 310 Area of a hexagon, 144 a regular n-gon, 146 a triangular lot, 310 a triangle, 306 an isosceles triangle, 144 Average walking speed in a city, 474, 502 Conversion functions, 79, 92 Depth of a submarine, 309 Dimensions of a rectangle, 13 triangular plot of land, 300 Distance between houses, 301 Distance to a plane, 310 an aircraft carrier, 299 a control tower, 191 a fire, 300 a hot air ballon, 299 a radar station, 140 Distance across a lake, 144, 310 marsh, 143 ravine, 450 Height of a building, 143, 145, 213, 296 hill, 300 kite, 300 paddle, 201 tree, 139, 213, 215 tower, 145 Motion of a point, 437 Number of diagonals of a polygon, 14

Perimeter of a regular n-gon, 146 Volume of water in a box, 49 cone, 50, 78 cylindrical tank, 266 Wingspan of an airplane, 309

• Music Musical scales, 465 Period of combined musical sound tracks, 214 Sound waves and beats, 202, 254

• Physics Angular rotation and angular speed, 131 Atmospheric pressure, 531 Average velocity, 71 Brachistochrone problem, 437 Carbon dating, 508, 509, 540 Current in a circuit, 518 Damped harmonic motion, 206, 209, 213 Decibel level, 491 Distance an object falls in a given time, 31, 504, 519 Distance a ball rolls down a ramp in a given time, 79, 107 Fermat’s Principle, 301 Frequency of a sound wave, 178 Hanging cable, 504 Intensity of light/sound, 464 Mach numbers, 245 Newton’s Law of Cooling, 518 Oscilloscope and sound waves, 178 Pendulum, 31, 534 Projectiles, 14, 273, 274, 279, 439, 447 Pulleys, 127, 130, 131 Radiation, 463 Radioactive decay, 508, 516, 544 Rocket launch, 190 Rotation versus lift distance of a winch, 131 Simple harmonic motion, 205, 206, 208, 209, 213 Snell’s Law, 301 Sonic boom, 401 Temperature of a fluid, 459, 465, 533 Time of descent, 113, 143 Velocity, 504, 515, 519 Voltage and amperage, 201 Water waves, 402 Work, 324, 326, 327, 330, 361

• Psychology and Learning Exposure to a rumor, 518 Learning theory, 519 Training/test scores, 503 Typing speed, 477, 502

• Sports and Recreation

• Technology and Engineering

Baseball bat speed, 104 Dimensions of a baseball field, 34 Distance from a golf tee to the pin, 299 Distance of a drive by a golfer, 299 Distance to first base, 309 Exercise heart rate, 502 Focus of a parabolic microphone, 372 Launch angle of a basketball, 288 Navigation of a boat, 505 Number of professional sports teams in a city, 95, 98 Olympic events, 526, 533 Parabolic ski design, 373 Scuba diving and hypothermia, 530, 531 Sky diving, 499 Speed of a boat, 104 Speed of a dragster, 502 Towing a boat, 78 Width of a skateboard jump ramp, 143 World records, 521, 531, 543

Angle between vectors, 322, 361 Angular speed of a wheel, 214 Direct-current motors, 105 Elliptical gears, 388 Engine design, 310 Focus of a satellite dish, 369, 372 Forces in equilibrium, 326 Hyperbolic gear, 401 Linear speed of a point on a rotating wheel, 214, 450 Logarithmic decrement of damped harmonic motion, 209 LORAN, 397, 400 Magnitude of a force, 319, 325, 326 Reflective property of a hyperbola, 398 Reflective property of a parabola, 369 Reflective property of an ellipse, 384 Rotary engine design, 447 Scotch yoke, 201

Structural defects, 371 Tones on a touch-tone phone, 252, 253 Uniform motion simulations, 441

• Travel Bearing of a boat, 303 Distance and bearing from a starting point, 310 Distance between a ship and a lighthouse, 45, 297, 300 Distance between airports, 300, 309 Distance between ships, 50, 310 Distance of a great circle route, 331 Great circle routes, 330 Ground speed/course of an airplane, 318, 325, 330, 333, 544 Heading of a boat, 325, 333 Initial heading required to fly a great circle route, 331 Lighthouse beacon, 201 Nautical miles and statute miles, 132

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Important Formulas Leg a

Pythagorean Theorem c 2 苷 a2 b2

Hypotenuse c

Leg b

The distance between P1共x1, y1兲 and P2共x2 , y2兲 is

Vertical and Horizontal Translations If f is a function and c is a positive constant, then the graph of ●

●

d共P1, P2兲苷兹共x2 x1兲2 共 y2 y1兲2 The slope m of a line through P1共x1, y1兲 and P2共x2, y2兲 is m苷

y2 y1 , x2 x1

x1 苷 x2

Quadratic Formula If a 苷 0, the solutions of ax 2 bx c 苷 0 are x苷

b 兹b2 4ac 2a

Properties of Functions A function is a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates. If a and b are elements of an interval I that is a subset of the domain of a function f, then ●

●

●

f is an increasing function on I if f共a兲 f共b兲 whenever a b. f is a decreasing function on I if f共a兲 f共b兲 whenever a b. f is a constant function on I if f共a兲苷 f共b兲 for all a and b.

A one-to-one function satisfies the additional condition that given any y, there is one and only one x that can be paired with that given y.

Graphing Concepts Odd Functions A function f is an odd function if f共x兲苷 f共x兲 for all x in the domain of f. The graph of an odd function is symmetric with respect to the origin. Even Functions A function is an even function if f共x兲苷 f共x兲 for all x in the domain of f. The graph of an even function is symmetric with respect to the y-axis.

●

●

y 苷 f共x兲 c is the graph of y 苷 f共x兲 shifted up vertically c units. y 苷 f共x兲 c is the graph of y 苷 f共x兲 shifted down vertically c units. y 苷 f共x c兲 is the graph of y 苷 f共x兲 shifted left horizontally c units. y 苷 f共x c兲 is the graph of y 苷 f共x兲 shifted right horizontally c units.

Reflections If f is a function then the graph of ●

●

y 苷 f共x兲 is the graph of y 苷 f共x兲 reflected across the x-axis. y 苷 f共x兲 is the graph of y 苷 f共x兲 reflected across the y-axis.

Vertical Shrinking and Stretching ● If c 0 and the graph of y 苷 f共x兲 contains the point 共x, y兲, then the graph of y 苷 c f共x兲 contains the point 共x, cy兲. ● If c 1, the graph of y 苷 c f共x兲 is obtained by stretching the graph of y 苷 f共x兲 away from the x-axis by a factor of c. ● If 0 c 1, the graph of y 苷 c f共x兲 is obtained by shrinking the graph of y 苷 f共x兲 toward the x-axis by a factor of c. Horizontal Shrinking and Stretching If a 0 and the graph of y 苷 f共x兲 contains the point 共x, y兲, then the graph of y 苷 f共ax兲 contains the 1 point x, y . a ● If a 1, the graph of y 苷 f共ax兲 is a horizontal shrinking of the graph of y 苷 f共x兲. ● If 0 a 1, the graph of y 苷 f共ax兲 is a horizontal stretching of the graph of y 苷 f共x兲. ●

冉冊

Definitions of Trigonometric Functions sin 苷

b r

csc 苷

r b

cos 苷

a r

sec 苷

r a

b a

cot 苷

tan 苷

where r 苷兹a2 b 2

a b

y

P(a, b) θ

b a

x

Definitions of Circular Functions sin t 苷 y

csc t 苷

1 y

Sum of Two Angle Identities sin 共兲苷 sin cos cos sin

y

1 x

cos t 苷 x

sec t 苷

y tan t 苷 x

x cot t 苷 y

cos 共兲苷 cos cos sin sin

t

P(x, y)

r=1

A(1, 0) x

tan tan 1 tan tan

tan 共兲苷

Difference of Two Angle Identities sin 共兲苷 sin cos cos sin cos 共兲苷 cos cos sin sin

Formulas for Triangles

tan tan 1 tan tan

For any triangle ABC, the following formulas can be used.

tan 共兲苷

Law of Sines a b c 苷苷 sin A sin B sin C

Double-Angle Identities

C

sin 2 苷 2 sin cos

Law of Cosines c 2 苷 a 2 b 2 2ab cos C Area of a Triangle 1 K 苷 ab sin C 2

a

b

A

B

c

cos 2 苷 cos2 sin2 苷 1 2 sin2 苷 2 cos2 1 tan 2 苷

K苷

a2 sin B sin C 2 sin A

Power-Reducing Identities

Heron’s Formula abc K 苷兹s共s a兲共s b兲共s c兲, where s 苷 2

sin2 苷

1 cos 2 2

cos2 苷

1 cos 2 2

tan2 苷

1 cos 2 1 cos 2

Fundamental Identities tan 苷

sin cos

sin2 cos2 苷 1

cot 苷

cos sin

Half-Angle Identities

1 tan2 苷 sec2

1 cot 2 苷 csc2

cos 共兲苷 cos

tan 共兲苷 tan

Reciprocal Identities csc 苷

1 sin

sec 苷

1 cos

cot 苷

1 tan

冑冑

sin

苷

2

1 cos 2

cos

苷

2

1 cos 2

tan

sin 1 cos 苷苷 2 1 cos sin

Formulas for Negatives sin 共兲苷 sin

2 tan 1 tan2

Cofunction Formulas sin tan

冉冊冉冊

苷 cos 2

cos

Sum-to-Product Identities

冉冊

苷 sin 2

苷 cot 2

Product-to-Sum Identities

sin x sin y 苷 2 sin

xy xy cos 2 2

sin x sin y 苷 2 cos

xy xy sin 2 2

cos x cos y 苷 2 cos

xy xy cos 2 2

2 sin cos 苷 sin 共兲 sin 共兲

cos x cos y 苷 2 sin

2 cos sin 苷 sin 共兲 sin 共兲

xy xy sin 2 2

2 cos cos 苷 cos 共兲 cos 共兲 2 sin sin 苷 cos 共兲 cos 共兲

The Trigonometric Functions Sine function

Cosine function

Tangent function y

y 1

−2π

−π

y f(x) = sin x

π

2π x

1

−2π

−π

−1

1

f(x) = cos x

π

2π x

− 2π

−π

π

2π x

−1

−1 f(x) = tan x

Cosecant function

−2π

Secant function

Cotangent function

y

y

y

1

1

1

−π

π −1

f (x) = csc x

2π x

−2π

−π

π −1

f(x) = sec x

2π x

− 2π

−π

π −1

f(x) = cot x

2π x